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<title>Reflexive space</title>

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Reflexive space

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In the area of mathematics known as <a href="Functional_analysis" title="Functional analysis">functional analysis</a>, a reflexive space is a <a href="Locally_convex" class="mw-redirect" title="Locally convex">locally convex</a> <a href="Topological_vector_space" title="Topological vector space">topological vector space</a> for which the canonical evaluation map from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> into its <a href="Bidual" class="mw-redirect" title="Bidual">bidual</a> (which is the <a href="Strong_dual" class="mw-redirect" title="Strong dual">strong dual</a> of the strong dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">) is a <a href="Homeomorphism" title="Homeomorphism">homeomorphism</a> (or equivalently, a <a href="TVS_isomorphism" class="mw-redirect" title="TVS isomorphism">TVS isomorphism</a>).

A <a href="Normed_space" class="mw-redirect" title="Normed space">normed space</a> is reflexive if and only if this canonical evaluation map is <a href="Surjective" class="mw-redirect" title="Surjective">surjective</a>, in which case this (always linear) evaluation map is an <a href="Isometric_isomorphism" class="mw-redirect" title="Isometric isomorphism">isometric isomorphism</a> and the normed space is a <a href="Banach_space" title="Banach space">Banach space</a>. Those spaces for which the canonical evaluation map is surjective are called <a href="Semi-reflexive" class="mw-redirect" title="Semi-reflexive">semi-reflexive</a> spaces.

In 1951, <a href="Robert_C._James" title="Robert C. James">R. C. James</a> discovered a Banach space, now known as <a href="James'_space" title="James' space">James' space</a>, that is not reflexive (meaning that the canonical evaluation map is not an isomorphism) but is nevertheless isometrically isomorphic to its bidual (any such <a href="Isometric_isomorphism" class="mw-redirect" title="Isometric isomorphism">isometric isomorphism</a> is necessarily not the canonical evaluation map). So importantly, for a Banach space to be reflexive, it is not enough for it to be isometrically isomorphic to its bidual; it is the canonical evaluation map in particular that has to be a homeomorphism.

Reflexive spaces play an important role in the general theory of <a href="Locally_convex" class="mw-redirect" title="Locally convex">locally convex</a> TVSs and in the theory of <a href="Banach_space" title="Banach space">Banach spaces</a> in particular. <a href="Hilbert_space" title="Hilbert space">Hilbert spaces</a> are prominent examples of reflexive Banach spaces. Reflexive Banach spaces are often characterized by their geometric properties.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<dl><dt>Definition of the bidual</dt></dl>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Bidual" class="mw-redirect" title="Bidual">Bidual</a></div>

Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a <a href="Topological_vector_space" title="Topological vector space">topological vector space</a> (TVS) over the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} }</annotation>

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</math><img src="./573f72afae7df709959ab1a58cd643743466a187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {F} }" loading="lazy"> (which is either the real or complex numbers) whose <a href="Continuous_dual_space" class="mw-redirect" title="Continuous dual space">continuous dual space</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime },}">

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<annotation encoding="application/x-tex">{\displaystyle X^{\prime },}</annotation>

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</math><img src="./90fef41e6ae66cd91995ce221cc99d754e3dd4ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.328ex; height:2.843ex;" alt="{\displaystyle X^{\prime },}" loading="lazy"> separates points on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (that is, for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,x\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X,x\neq 0}</annotation>

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</math><img src="./57df604f721bc429ab89cfe87049a6c4611e00af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.775ex; height:2.676ex;" alt="{\displaystyle x\in X,x\neq 0}" loading="lazy"> there exists some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\prime }\in X^{\prime }}">

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<annotation encoding="application/x-tex">{\displaystyle x^{\prime }\in X^{\prime }}</annotation>

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</math><img src="./f7e2705801df172352ed10c146e579c0338915a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.537ex; height:2.509ex;" alt="{\displaystyle x^{\prime }\in X^{\prime }}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\prime }(x)\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle x^{\prime }(x)\neq 0}</annotation>

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</math><img src="./f4f96d180ed26037f5174ea832d65537cee7e5e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.414ex; height:3.009ex;" alt="{\displaystyle x^{\prime }(x)\neq 0}" loading="lazy">).

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{b}^{\prime }}">

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<annotation encoding="application/x-tex">{\displaystyle X_{b}^{\prime }}</annotation>

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</math><img src="./1852874a49781f1d80418a3946541499736a9611.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.862ex; height:2.843ex;" alt="{\displaystyle X_{b}^{\prime }}" loading="lazy"> (some texts write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\beta }^{\prime }}">

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<annotation encoding="application/x-tex">{\displaystyle X_{\beta }^{\prime }}</annotation>

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</math><img src="./a9afb4be9e601adc17a887dc64d9bd792c282d3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.098ex; height:3.176ex;" alt="{\displaystyle X_{\beta }^{\prime }}" loading="lazy">) denote the <a href="Strong_dual" class="mw-redirect" title="Strong dual">strong dual</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

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<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

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</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> which is the vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime }}">

<msup>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X^{\prime }}</annotation>

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</math><img src="./d7a3a5819cc45f097de14b3ac5a8bedd902bc66d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X^{\prime }}" loading="lazy"> of continuous linear functionals on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> endowed with the <a href="Topology_of_uniform_convergence" class="mw-redirect" title="Topology of uniform convergence">topology of uniform convergence</a> on <a href="Topology_of_uniform_convergence" class="mw-redirect" title="Topology of uniform convergence">bounded subsets</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">;

this topology is also called the strong dual topology and it is the "default" topology placed on a continuous dual space (unless another topology is specified).

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a normed space, then the strong dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is the continuous dual space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime }}">

<msup>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X^{\prime }}</annotation>

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</math><img src="./d7a3a5819cc45f097de14b3ac5a8bedd902bc66d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X^{\prime }}" loading="lazy"> with its usual norm topology.

The bidual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

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<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

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</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime \prime },}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle X^{\prime \prime },}</annotation>

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</math><img src="./d90418bd6f0aa183515366ab76544d0c5217b1c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.781ex; height:2.843ex;" alt="{\displaystyle X^{\prime \prime },}" loading="lazy"> is the strong dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{b}^{\prime }}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X_{b}^{\prime }}</annotation>

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</math><img src="./1852874a49781f1d80418a3946541499736a9611.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.862ex; height:2.843ex;" alt="{\displaystyle X_{b}^{\prime }}" loading="lazy">; that is, it is the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime }.}">

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<annotation encoding="application/x-tex">{\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime }.}</annotation>

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</math><img src="./dc43396e27ef936ef20eb0c75201081989d671da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.576ex; height:3.676ex;" alt="{\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime }.}" loading="lazy"><a href="#cite_note-FOOTNOTETrèves2006372–374-1">[1]</a>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a normed space, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime \prime }}">

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X^{\prime \prime }}</annotation>

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</math><img src="./9f905bf6c4a877c966241e9d7650d10e3978b3dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.134ex; height:2.509ex;" alt="{\displaystyle X^{\prime \prime }}" loading="lazy"> is the continuous dual space of the Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{b}^{\prime }}">

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<annotation encoding="application/x-tex">{\displaystyle X_{b}^{\prime }}</annotation>

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</math><img src="./1852874a49781f1d80418a3946541499736a9611.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.862ex; height:2.843ex;" alt="{\displaystyle X_{b}^{\prime }}" loading="lazy"> with its usual norm topology.

<dl><dt>Definitions of the evaluation map and reflexive spaces</dt></dl>

For any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation>

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</math><img src="./e3ebdb0a09f0721ccdd0b779e0a21caf386be82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" loading="lazy"> let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{x}:X^{\prime }\to \mathbb {F} }">

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<annotation encoding="application/x-tex">{\displaystyle J_{x}:X^{\prime }\to \mathbb {F} }</annotation>

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</math><img src="./703cb77c4b2b31c49a9a95bc7b4158f1701680cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.116ex; height:2.843ex;" alt="{\displaystyle J_{x}:X^{\prime }\to \mathbb {F} }" loading="lazy"> be defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{x}\left(x^{\prime }\right)=x^{\prime }(x),}">

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle J_{x}\left(x^{\prime }\right)=x^{\prime }(x),}</annotation>

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</math><img src="./f63d2f52c2441f7f27d97d5f93900a2cdca2bfe1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.572ex; height:3.009ex;" alt="{\displaystyle J_{x}\left(x^{\prime }\right)=x^{\prime }(x),}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{x}}">

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<annotation encoding="application/x-tex">{\displaystyle J_{x}}</annotation>

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</math><img src="./573c4eac7701ac45330a1677abe1c3a7fc054ae1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.463ex; height:2.509ex;" alt="{\displaystyle J_{x}}" loading="lazy"> is a linear map called the evaluation map at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">;

since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{x}:X_{b}^{\prime }\to \mathbb {F} }">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle J_{x}:X_{b}^{\prime }\to \mathbb {F} }</annotation>

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</math><img src="./fa75e66c41fcec3a4a7d1a84f807f7a6e8607825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.296ex; height:2.843ex;" alt="{\displaystyle J_{x}:X_{b}^{\prime }\to \mathbb {F} }" loading="lazy"> is necessarily continuous, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{x}\in \left(X_{b}^{\prime }\right)^{\prime }.}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle J_{x}\in \left(X_{b}^{\prime }\right)^{\prime }.}</annotation>

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</math><img src="./64b8859bb94bb2aaaaa48c4870c7696b7bdf2017.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.627ex; height:3.509ex;" alt="{\displaystyle J_{x}\in \left(X_{b}^{\prime }\right)^{\prime }.}" loading="lazy">

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime }}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle X^{\prime }}</annotation>

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</math><img src="./d7a3a5819cc45f097de14b3ac5a8bedd902bc66d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X^{\prime }}" loading="lazy"> separates points on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

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</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> the linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J:X\to \left(X_{b}^{\prime }\right)^{\prime }}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle J:X\to \left(X_{b}^{\prime }\right)^{\prime }}</annotation>

</semantics>

</math><img src="./bd05513fb4e4f11626375d0093acf0dac174836f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.679ex; height:3.509ex;" alt="{\displaystyle J:X\to \left(X_{b}^{\prime }\right)^{\prime }}" loading="lazy"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(x):=J_{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(x):=J_{x}}</annotation>

</semantics>

</math><img src="./cc13235309217d28d014c5b4b40fc21559bf578c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.818ex; height:2.843ex;" alt="{\displaystyle J(x):=J_{x}}" loading="lazy"> is injective where this map is called the evaluation map or the canonical map.

Call <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> <a href="Semi-reflexive" class="mw-redirect" title="Semi-reflexive">semi-reflexive</a> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J:X\to \left(X_{b}^{\prime }\right)^{\prime }}">

<msup>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to \left(X_{b}^{\prime }\right)^{\prime }}</annotation>

</semantics>

</math><img src="./bd05513fb4e4f11626375d0093acf0dac174836f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.679ex; height:3.509ex;" alt="{\displaystyle J:X\to \left(X_{b}^{\prime }\right)^{\prime }}" loading="lazy"> is bijective (or equivalently, <a href="Surjective" class="mw-redirect" title="Surjective">surjective</a>) and we call <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> reflexive if in addition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J:X\to X^{\prime \prime }=\left(X_{b}^{\prime }\right)_{b}^{\prime }}">

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to X^{\prime \prime }=\left(X_{b}^{\prime }\right)_{b}^{\prime }}</annotation>

</semantics>

</math><img src="./8556a3e03e3d30c27fd73ce788f57c9373a5a138.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.164ex; height:3.676ex;" alt="{\displaystyle J:X\to X^{\prime \prime }=\left(X_{b}^{\prime }\right)_{b}^{\prime }}" loading="lazy"> is an isomorphism of TVSs.<a href="#cite_note-FOOTNOTETrèves2006372–374-1">[1]</a>

A <a href="Normable" class="mw-redirect" title="Normable">normable</a> space is reflexive if and only if it is semi-reflexive or equivalently, if and only if the evaluation map is surjective.

<div class="mw-heading mw-heading2"><h2 id="Reflexive_Banach_spaces">Reflexive Banach spaces</h2></div>

Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a <a href="Normed_vector_space" title="Normed vector space">normed vector space</a> over the number field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} =\mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} =\mathbb {R} }</annotation>

</semantics>

</math><img src="./f9e1200c55f062d38108d482407e8a36b868eb91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {F} =\mathbb {R} }" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} =\mathbb {C} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} =\mathbb {C} }</annotation>

</semantics>

</math><img src="./72f4a6d534b95f56864003cec22d143b2361eba9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {F} =\mathbb {C} }" loading="lazy"> (the <a href="Real_number" title="Real number">real numbers</a> or the <a href="Complex_number" title="Complex number">complex numbers</a>), with a norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\,\cdot \,\|.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\,\cdot \,\|.}</annotation>

</semantics>

</math><img src="./ba41443b931e730f5c17a10a2c3363b292c918f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.425ex; height:2.843ex;" alt="{\displaystyle \|\,\cdot \,\|.}" loading="lazy"> Consider its <a href="Dual_norm" title="Dual norm">dual normed space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime },}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime },}</annotation>

</semantics>

</math><img src="./90fef41e6ae66cd91995ce221cc99d754e3dd4ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.328ex; height:2.843ex;" alt="{\displaystyle X^{\prime },}" loading="lazy"> that consists of all <a href="Continuous_function" title="Continuous function">continuous</a> <a href="Linear_functional" class="mw-redirect" title="Linear functional">linear functionals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to \mathbb {F} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to \mathbb {F} }</annotation>

</semantics>

</math><img src="./3a8a1718f96d2610a6fe230d7533cf2f4437fa27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.23ex; height:2.509ex;" alt="{\displaystyle f:X\to \mathbb {F} }" loading="lazy"> and is equipped with the <a href="Dual_norm" title="Dual norm">dual norm</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\,\cdot \,\|^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\,\cdot \,\|^{\prime }}</annotation>

</semantics>

</math><img src="./2d8975cc0aef88891315d67cd27175244eb1edf1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:3.009ex;" alt="{\displaystyle \|\,\cdot \,\|^{\prime }}" loading="lazy"> defined by

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|f\|^{\prime }=\sup\{|f(x)|\,:\,x\in X,\ \|x\|=1\}.}</annotation>

</semantics>

</math>

The dual <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime }}</annotation>

</semantics>

</math><img src="./d7a3a5819cc45f097de14b3ac5a8bedd902bc66d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X^{\prime }}" loading="lazy"> is a normed space (a <a href="Banach_space" title="Banach space">Banach space</a> to be precise), and its dual normed space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime \prime }=\left(X^{\prime }\right)^{\prime }}">

<msup>

</mrow>

</msup>

<msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime \prime }=\left(X^{\prime }\right)^{\prime }}</annotation>

</semantics>

</math><img src="./af02204ab7cc04eed33c8ec52dd2f7372cba03b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.408ex; height:3.343ex;" alt="{\displaystyle X^{\prime \prime }=\left(X^{\prime }\right)^{\prime }}" loading="lazy"> is called bidual space for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"> The bidual consists of all continuous linear functionals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:X^{\prime }\to \mathbb {F} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h:X^{\prime }\to \mathbb {F} }</annotation>

</semantics>

</math><img src="./922e403101551010ed01e5ab6112a2a852827b7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.992ex; height:2.509ex;" alt="{\displaystyle h:X^{\prime }\to \mathbb {F} }" loading="lazy"> and is equipped with the norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\,\cdot \,\|^{\prime \prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\,\cdot \,\|^{\prime \prime }}</annotation>

</semantics>

</math><img src="./5257c9c5eb37b7cf909df75c1d9064a5d1d36801.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.915ex; height:3.009ex;" alt="{\displaystyle \|\,\cdot \,\|^{\prime \prime }}" loading="lazy"> dual to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\,\cdot \,\|^{\prime }.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\,\cdot \,\|^{\prime }.}</annotation>

</semantics>

</math><img src="./47cf778586585b0f9a1dbed504079331ec1b76d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.11ex; height:3.009ex;" alt="{\displaystyle \|\,\cdot \,\|^{\prime }.}" loading="lazy"> Each vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> generates a scalar function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(x):X^{\prime }\to \mathbb {F} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(x):X^{\prime }\to \mathbb {F} }</annotation>

</semantics>

</math><img src="./35455ab244ad7bc25b6b3ac92669474a88e506d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.263ex; height:3.009ex;" alt="{\displaystyle J(x):X^{\prime }\to \mathbb {F} }" loading="lazy"> by the formula:

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(x)(f)=f(x)\qquad {\text{ for all }}f\in X^{\prime },}</annotation>

</semantics>

</math>

and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(x)}</annotation>

</semantics>

</math><img src="./40fcaf0063846d508ceeb126a7a2bc1b0229a0ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.61ex; height:2.843ex;" alt="{\displaystyle J(x)}" loading="lazy"> is a continuous linear functional on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime },}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime },}</annotation>

</semantics>

</math><img src="./90fef41e6ae66cd91995ce221cc99d754e3dd4ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.328ex; height:2.843ex;" alt="{\displaystyle X^{\prime },}" loading="lazy"> that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(x)\in X^{\prime \prime }.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(x)\in X^{\prime \prime }.}</annotation>

</semantics>

</math><img src="./1ce0890140eb809cd8b6d93ab31b483674a76304.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.232ex; height:3.009ex;" alt="{\displaystyle J(x)\in X^{\prime \prime }.}" loading="lazy"> One obtains in this way a map

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to X^{\prime \prime }}</annotation>

</semantics>

</math>

called evaluation map, that is linear. It follows from the <a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach theorem</a> that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

</semantics>

</math><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> is injective and preserves norms:

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{ for all }}x\in X\qquad \|J(x)\|^{\prime \prime }=\|x\|,}</annotation>

</semantics>

</math>

that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

</semantics>

</math><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> isometrically onto its image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(X)}</annotation>

</semantics>

</math><img src="./c1ec6aa47953554d0e553013321bdbbc3dab105a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.261ex; height:2.843ex;" alt="{\displaystyle J(X)}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime \prime }.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime \prime }.}</annotation>

</semantics>

</math><img src="./cfae32fc5d9dfc08280fa75d84ce313cfd603cb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.781ex; height:2.509ex;" alt="{\displaystyle X^{\prime \prime }.}" loading="lazy"> Furthermore, the image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(X)}</annotation>

</semantics>

</math><img src="./c1ec6aa47953554d0e553013321bdbbc3dab105a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.261ex; height:2.843ex;" alt="{\displaystyle J(X)}" loading="lazy"> is closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime \prime },}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime \prime },}</annotation>

</semantics>

</math><img src="./d90418bd6f0aa183515366ab76544d0c5217b1c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.781ex; height:2.843ex;" alt="{\displaystyle X^{\prime \prime },}" loading="lazy"> but it need not be equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime \prime }.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime \prime }.}</annotation>

</semantics>

</math><img src="./cfae32fc5d9dfc08280fa75d84ce313cfd603cb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.781ex; height:2.509ex;" alt="{\displaystyle X^{\prime \prime }.}" loading="lazy">

A normed space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is called reflexive if it satisfies the following equivalent conditions:

<li> the evaluation map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J:X\to X^{\prime \prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to X^{\prime \prime }}</annotation>

</semantics>

</math><img src="./c51ba0fa314c691e9410e8807b52f4a8dc98fc6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.137ex; height:2.509ex;" alt="{\displaystyle J:X\to X^{\prime \prime }}" loading="lazy"> is <a href="Bijection%2C_injection_and_surjection" title="Bijection, injection and surjection">surjective</a>,</li>

<li> the evaluation map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J:X\to X^{\prime \prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to X^{\prime \prime }}</annotation>

</semantics>

</math><img src="./c51ba0fa314c691e9410e8807b52f4a8dc98fc6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.137ex; height:2.509ex;" alt="{\displaystyle J:X\to X^{\prime \prime }}" loading="lazy"> is an <a href="Banach_space" title="Banach space">isometric isomorphism</a> of normed spaces,</li>

<li> the evaluation map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J:X\to X^{\prime \prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to X^{\prime \prime }}</annotation>

</semantics>

</math><img src="./c51ba0fa314c691e9410e8807b52f4a8dc98fc6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.137ex; height:2.509ex;" alt="{\displaystyle J:X\to X^{\prime \prime }}" loading="lazy"> is an <a href="Banach_space" title="Banach space">isomorphism</a> of normed spaces.</li>

</ol>

A reflexive space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a Banach space, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is then isometric to the Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime \prime }.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime \prime }.}</annotation>

</semantics>

</math><img src="./cfae32fc5d9dfc08280fa75d84ce313cfd603cb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.781ex; height:2.509ex;" alt="{\displaystyle X^{\prime \prime }.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Remark">Remark</h3></div>

A Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is reflexive if it is linearly isometric to its bidual under this canonical embedding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J.}</annotation>

</semantics>

</math><img src="./0e11c56789d11402ffe7a3b44a770476d3c7d4d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.118ex; height:2.176ex;" alt="{\displaystyle J.}" loading="lazy"> <a href="James'_space" title="James' space">James' space</a> is an example of a non-reflexive space which is linearly isometric to its <a href="Dual_space#Double_dual" title="Dual space">bidual</a>. Furthermore, the image of James' space under the canonical embedding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

</semantics>

</math><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> has <a href="Codimension" title="Codimension">codimension</a> one in its bidual.

A Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is called quasi-reflexive (of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">) if the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime \prime }/J(X)}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime \prime }/J(X)}</annotation>

</semantics>

</math><img src="./a9fc11ce9eeb5e4b103d19ea041c506b01400fc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.557ex; height:3.009ex;" alt="{\displaystyle X^{\prime \prime }/J(X)}" loading="lazy"> has finite dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d.}</annotation>

</semantics>

</math><img src="./a6636c10463ecb6f29e5bb5c3a7512265add0369.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.863ex; height:2.176ex;" alt="{\displaystyle d.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

<ol><li>Every finite-dimensional normed space is reflexive, simply because in this case, the space, its dual and bidual all have the same linear dimension, hence the linear injection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

</semantics>

</math><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> from the definition is bijective, by the <a href="Rank%E2%80%93nullity_theorem" title="Rank–nullity theorem">rank–nullity theorem</a>.</li>

<li>The Banach space <a href="Sequence_space#c_and_c0" title="Sequence space"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{0}}</annotation>

</semantics>

</math><img src="./1882ba8f1dc60f0c68a642abb5af093c73910921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}" loading="lazy"></a> of scalar sequences tending to 0 at infinity, equipped with the supremum norm, is not reflexive. It follows from the general properties below that <a href="Sequence_space#.E2.84.93p_spaces" title="Sequence space"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{1}}</annotation>

</semantics>

</math><img src="./7d1156e1c2220628042b0fc51e0c73deb3b7c6d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation>

</semantics>

</math><img src="./8348195cf09473662c6f59e6717722a6fc01d0f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}" loading="lazy"></a> are not reflexive, because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{1}}</annotation>

</semantics>

</math><img src="./7d1156e1c2220628042b0fc51e0c73deb3b7c6d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{1}}" loading="lazy"> is isomorphic to the dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{0}}</annotation>

</semantics>

</math><img src="./1882ba8f1dc60f0c68a642abb5af093c73910921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation>

</semantics>

</math><img src="./8348195cf09473662c6f59e6717722a6fc01d0f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}" loading="lazy"> is isomorphic to the dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{1}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{1}.}</annotation>

</semantics>

</math><img src="./615f8bbfccb6efabd743f484610f32c6898a5a98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.671ex; height:2.676ex;" alt="{\displaystyle \ell ^{1}.}" loading="lazy"></li>

<li>All <a href="Hilbert_space" title="Hilbert space">Hilbert spaces</a> are reflexive, as are the <a href="Lp_space" title="Lp space">Lp spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation>

</semantics>

</math><img src="./cf2317aaca1ecee4b8ccf667bc1001059eae5850.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<p<\infty .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1<p<\infty .}</annotation>

</semantics>

</math><img src="./2d08398e23489426d42bcc4110899e4022b338b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.499ex; height:2.509ex;" alt="{\displaystyle 1<p<\infty .}" loading="lazy"> More generally: all <a href="Uniformly_convex_space" title="Uniformly convex space">uniformly convex</a> Banach spaces are reflexive according to the <a href="Milman%E2%80%93Pettis_theorem" title="Milman–Pettis theorem">Milman–Pettis theorem</a>. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}(\mu )}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{1}(\mu )}</annotation>

</semantics>

</math><img src="./b81a45f5ed33188fc3e64cd079882bbc7de34c24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.848ex; height:3.176ex;" alt="{\displaystyle L^{1}(\mu )}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }(\mu )}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{\infty }(\mu )}</annotation>

</semantics>

</math><img src="./5b7867eb72dc22e91568af1af857fd364f42458c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.669ex; height:2.843ex;" alt="{\displaystyle L^{\infty }(\mu )}" loading="lazy"> spaces are not reflexive (unless they are finite dimensional, which happens for example when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> is a measure on a finite set). Likewise, the Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C([0,1])}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C([0,1])}</annotation>

</semantics>

</math><img src="./44211c4c325ea7edb9462e7ccecda09841a41216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.228ex; height:2.843ex;" alt="{\displaystyle C([0,1])}" loading="lazy"> of continuous functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

</semantics>

</math><img src="./738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy"> is not reflexive.</li>

<li>The spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{p}(H)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{p}(H)}</annotation>

</semantics>

</math><img src="./2f5b7ecee3b4f83e2d8456427f2e423c0f91d218.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.357ex; height:3.009ex;" alt="{\displaystyle S_{p}(H)}" loading="lazy"> of operators in the <a href="Schatten_class_operator" title="Schatten class operator">Schatten class</a> on a Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> are uniformly convex, hence reflexive, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<p<\infty .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1<p<\infty .}</annotation>

</semantics>

</math><img src="./2d08398e23489426d42bcc4110899e4022b338b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.499ex; height:2.509ex;" alt="{\displaystyle 1<p<\infty .}" loading="lazy"> When the dimension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> is infinite, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}(H)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{1}(H)}</annotation>

</semantics>

</math><img src="./cf8be2e7e462b5cb2699496b6032786a2a7bce2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.352ex; height:2.843ex;" alt="{\displaystyle S_{1}(H)}" loading="lazy"> (the <a href="Trace_class" title="Trace class">trace class</a>) is not reflexive, because it contains a subspace isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{1},}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{1},}</annotation>

</semantics>

</math><img src="./03ededdd80a2eaaf2c4b55f0effe594d389a7646.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.671ex; height:3.009ex;" alt="{\displaystyle \ell ^{1},}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\infty }(H)=L(H)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{\infty }(H)=L(H)}</annotation>

</semantics>

</math><img src="./6f94237148cdc2ae145905d925287d17923dea94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.727ex; height:2.843ex;" alt="{\displaystyle S_{\infty }(H)=L(H)}" loading="lazy"> (the bounded linear operators on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">) is not reflexive, because it contains a subspace isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }.}</annotation>

</semantics>

</math><img src="./2e024835e5ef8ddeccb2fd0863554f526dfafd5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.492ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }.}" loading="lazy"> In both cases, the subspace can be chosen to be the operators diagonal with respect to a given orthonormal basis of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H.}</annotation>

</semantics>

</math><img src="./8933ae7244305ae7824aa18e077d1cf946e2ee9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.71ex; height:2.176ex;" alt="{\displaystyle H.}" loading="lazy"></li></ol>

<div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3></div>

Since every finite-dimensional normed space is a reflexive <a href="Banach_space" title="Banach space">Banach space</a>, only infinite-dimensional spaces can be non-reflexive.

If a Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is isomorphic to a reflexive Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is reflexive.<a href="#cite_note-3">[3]</a>

Every <a href="Closed_set" title="Closed set">closed</a> <a href="Linear_subspace" title="Linear subspace">linear subspace</a> of a reflexive space is reflexive. The continuous dual of a reflexive space is reflexive. Every <a href="Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">quotient</a> of a reflexive space by a closed subspace is reflexive.<a href="#cite_note-4">[4]</a>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> be a Banach space. The following are equivalent.

<ol>

<li>The space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is reflexive.</li>

<li>The continuous dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is reflexive.<a href="#cite_note-5">[5]</a></li>

<li>The closed unit ball of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is <a href="Compact_space" title="Compact space">compact</a> in the <a href="Weak_topology" title="Weak topology">weak topology</a>. (This is known as Kakutani's Theorem.)<a href="#cite_note-FOOTNOTEConway1985Theorem_V.4.2,_p.&nbsp;135-6">[6]</a></li>

<li>Every bounded sequence in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> has a weakly convergent subsequence.<a href="#cite_note-7">[7]</a></li>

<li>The statement of <a href="Riesz's_lemma" title="Riesz's lemma">Riesz's lemma</a> holds when the real number<a href="#cite_note-8">[note 1]</a> is exactly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1.}</annotation>

</semantics>

</math><img src="./af8c4e445819b13a052647aa3eb2be990b0a4b24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 1.}" loading="lazy"><a href="#cite_note-FOOTNOTEDiestel19846-9">[8]</a> Explicitly, for every closed proper vector subspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> there exists some vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\in X}</annotation>

</semantics>

</math><img src="./4c358e942a980691c36011ffbfd456695180544e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle u\in X}" loading="lazy"> of unit norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u\|=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|u\|=1}</annotation>

</semantics>

</math><img src="./e1fb084f1df143ead8205a9abdab81e62067712e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.916ex; height:2.843ex;" alt="{\displaystyle \|u\|=1}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u-y\|\geq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|u-y\|\geq 1}</annotation>

</semantics>

</math><img src="./954ca80473fff1f44ae0dc999f7ffc63b25c64e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.911ex; height:2.843ex;" alt="{\displaystyle \|u-y\|\geq 1}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\in Y.}</annotation>

</semantics>

</math><img src="./b14bc7ecf2f86320b4a930f4961919ae45a34d9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.416ex; height:2.509ex;" alt="{\displaystyle y\in Y.}" loading="lazy">

<ul><li>Using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(u,Y):=\inf _{y\in Y}\|u-y\|}">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d(u,Y):=\inf _{y\in Y}\|u-y\|}</annotation>

</semantics>

</math><img src="./f08d7c51db9ef64999d4196b9bf1ea4cd42824a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.112ex; height:4.343ex;" alt="{\displaystyle d(u,Y):=\inf _{y\in Y}\|u-y\|}" loading="lazy"> to denote the distance between the vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> and the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y,}</annotation>

</semantics>

</math><img src="./a3765557b7effa1a5f2f4dce9c80a25973b7009f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" loading="lazy"> this can be restated in simpler language as: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is reflexive if and only if for every closed proper vector subspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y,}</annotation>

</semantics>

</math><img src="./a3765557b7effa1a5f2f4dce9c80a25973b7009f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" loading="lazy"> there is some vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> on the <a href="Unit_sphere" title="Unit sphere">unit sphere</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> that is always at least a distance of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=d(u,Y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1=d(u,Y)}</annotation>

</semantics>

</math><img src="./4bbe42f541c16cfcec106dc16bd85f8362034eea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.423ex; height:2.843ex;" alt="{\displaystyle 1=d(u,Y)}" loading="lazy"> away from the subspace.</li>

<li>For example, if the reflexive Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\mathbb {R} ^{3}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=\mathbb {R} ^{3}}</annotation>

</semantics>

</math><img src="./573ab31dabc14a9acb5775793f3b912617689c5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.811ex; height:2.676ex;" alt="{\displaystyle X=\mathbb {R} ^{3}}" loading="lazy"> is endowed with the usual <a href="Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\mathbb {R} \times \mathbb {R} \times \{0\}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=\mathbb {R} \times \mathbb {R} \times \{0\}}</annotation>

</semantics>

</math><img src="./7539843483a8586f3f60fb47144209876f29783f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.396ex; height:2.843ex;" alt="{\displaystyle Y=\mathbb {R} \times \mathbb {R} \times \{0\}}" loading="lazy"> is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x-y}</annotation>

</semantics>

</math><img src="./3129cb3620bd9f38d0304a0fca719644d7d2d265.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle x-y}" loading="lazy"> plane then the points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=(0,0,\pm 1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u=(0,0,\pm 1)}</annotation>

</semantics>

</math><img src="./851f0f2adadf047be2a9320028cdbc751b81d096.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.601ex; height:2.843ex;" alt="{\displaystyle u=(0,0,\pm 1)}" loading="lazy"> satisfy the conclusion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(u,Y)=1.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d(u,Y)=1.}</annotation>

</semantics>

</math><img src="./15ad4811d1ad7c917d46fdf41182c8e6a09657c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.07ex; height:2.843ex;" alt="{\displaystyle d(u,Y)=1.}" loading="lazy"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is instead the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">-axis then every point belonging to the unit circle in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x-y}</annotation>

</semantics>

</math><img src="./3129cb3620bd9f38d0304a0fca719644d7d2d265.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle x-y}" loading="lazy"> plane satisfies the conclusion.</li></ul>

</li>

<li>Every continuous linear functional on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> attains its supremum on the closed unit ball in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"><a href="#cite_note-10">[9]</a> (<a href="James'_theorem" class="mw-redirect" title="James' theorem">James' theorem</a>)</li>

</ol>

Since norm-closed <a href="Convex_set" title="Convex set">convex subsets</a> in a Banach space are weakly closed,<a href="#cite_note-11">[10]</a>

it follows from the third property that closed bounded convex subsets of a reflexive space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> are weakly compact. Thus, for every decreasing sequence of non-empty closed bounded convex subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> the intersection is non-empty. As a consequence, every continuous <a href="Convex_function" title="Convex function">convex function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> on a closed convex subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> such that the set

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{t}=\{x\in C\,:\,f(x)\leq t\}}</annotation>

</semantics>

</math>

is non-empty and bounded for some real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t,}</annotation>

</semantics>

</math><img src="./4ea3ad87830a1055c7b85c04cf940cfd3b847ae6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.486ex; height:2.343ex;" alt="{\displaystyle t,}" loading="lazy"> attains its minimum value on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C.}</annotation>

</semantics>

</math><img src="./067be67e68f60c53ce83241748d0d6249675c58d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.413ex; height:2.176ex;" alt="{\displaystyle C.}" loading="lazy">

The promised geometric property of reflexive Banach spaces is the following: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is a closed non-empty <a href="Convex_set" title="Convex set">convex</a> subset of the reflexive space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> then for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c\in C}</annotation>

</semantics>

</math><img src="./447d3982c94c23d6b6d01c90da81a6125aa26567.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.614ex; height:2.176ex;" alt="{\displaystyle c\in C}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x-c\|}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|x-c\|}</annotation>

</semantics>

</math><img src="./cdced8eee39a4aed21709b1ccb4c0e960edb534d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.502ex; height:2.843ex;" alt="{\displaystyle \|x-c\|}" loading="lazy"> minimizes the distance between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and points of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C.}</annotation>

</semantics>

</math><img src="./067be67e68f60c53ce83241748d0d6249675c58d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.413ex; height:2.176ex;" alt="{\displaystyle C.}" loading="lazy"> This follows from the preceding result for convex functions, applied to<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y)+\|y-x\|.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(y)+\|y-x\|.}</annotation>

</semantics>

</math><img src="./51734364194312a4046d907e37db87c1966cbbf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.381ex; height:2.843ex;" alt="{\displaystyle f(y)+\|y-x\|.}" loading="lazy"> Note that while the minimal distance between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is uniquely defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,}</annotation>

</semantics>

</math><img src="./feff4d40084c7351bf57b11ba2427f6331f5bdbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}" loading="lazy"> the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> is not. The closest point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> is unique when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is uniformly convex.

A reflexive Banach space is <a href="Separable_space" title="Separable space">separable</a> if and only if its continuous dual is separable. This follows from the fact that for every normed space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y,}</annotation>

</semantics>

</math><img src="./a3765557b7effa1a5f2f4dce9c80a25973b7009f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" loading="lazy"> separability of the continuous dual <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y^{\prime }}</annotation>

</semantics>

</math><img src="./dc6b95f963ad415c9e50171ecb1826cc4c094477.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.585ex; height:2.343ex;" alt="{\displaystyle Y^{\prime }}" loading="lazy"> implies separability of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y.}</annotation>

</semantics>

</math><img src="./0c668649af47a30006f93c9847d61fee8d9ffb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" loading="lazy"><a href="#cite_note-12">[11]</a>

<div class="mw-heading mw-heading3"><h3 id="Super-reflexive_space">Super-reflexive space</h3></div>

Informally, a super-reflexive Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> has the following property: given an arbitrary Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y,}</annotation>

</semantics>

</math><img src="./a3765557b7effa1a5f2f4dce9c80a25973b7009f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" loading="lazy"> if all finite-dimensional subspaces of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> have a very similar copy sitting somewhere in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> must be reflexive. By this definition, the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> itself must be reflexive. As an elementary example, every Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> whose two dimensional subspaces are <a href="Isometry" title="Isometry">isometric</a> to subspaces of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\ell ^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=\ell ^{2}}</annotation>

</semantics>

</math><img src="./c7695788a893dd7e6ebddf9caa77140f48133161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.102ex; height:2.676ex;" alt="{\displaystyle X=\ell ^{2}}" loading="lazy"> satisfies the <a href="Parallelogram_law" title="Parallelogram law">parallelogram law</a>, hence<a href="#cite_note-13">[12]</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is a Hilbert space, therefore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is reflexive. So <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}}</annotation>

</semantics>

</math><img src="./91f1f909abd70bd3d8fff0f7ae1ac23052387e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}}" loading="lazy"> is super-reflexive.

The formal definition does not use isometries, but almost isometries. A Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is finitely representable<a href="#cite_note-SRBS-14">[13]</a>

in a Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> if for every finite-dimensional subspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{0}}</annotation>

</semantics>

</math><img src="./d4305e31e20568c004e0e4c8540bbfe2730a42cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.405ex; height:2.509ex;" alt="{\displaystyle Y_{0}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> and every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon >0,}</annotation>

</semantics>

</math><img src="./44c08d32cc0a46cfa7ccabd48ba8a50a87e0ca66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.852ex; height:2.509ex;" alt="{\displaystyle \epsilon >0,}" loading="lazy"> there is a subspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{0}}</annotation>

</semantics>

</math><img src="./6381fdad2b9f11954b1fc2db08bbaccf634ededa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{0}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> such that the multiplicative <a href="Banach%E2%80%93Mazur_compactum" title="Banach–Mazur compactum">Banach–Mazur distance</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{0}}</annotation>

</semantics>

</math><img src="./6381fdad2b9f11954b1fc2db08bbaccf634ededa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{0}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{0}}</annotation>

</semantics>

</math><img src="./d4305e31e20568c004e0e4c8540bbfe2730a42cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.405ex; height:2.509ex;" alt="{\displaystyle Y_{0}}" loading="lazy"> satisfies

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\left(X_{0},Y_{0}\right)<1+\varepsilon .}</annotation>

</semantics>

</math>

A Banach space finitely representable in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}}</annotation>

</semantics>

</math><img src="./91f1f909abd70bd3d8fff0f7ae1ac23052387e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}}" loading="lazy"> is a Hilbert space. Every Banach space is finitely representable in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{0}.}</annotation>

</semantics>

</math><img src="./fb9fae09ed0e2be745188ad807c16b2570f52ff7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.708ex; height:2.009ex;" alt="{\displaystyle c_{0}.}" loading="lazy"> The <a href="Lp_space" title="Lp space">Lp space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}([0,1])}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{p}([0,1])}</annotation>

</semantics>

</math><img src="./d53d55ccc06ce5cc9ec3bede2be3e7933c206ee3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.104ex; height:2.843ex;" alt="{\displaystyle L^{p}([0,1])}" loading="lazy"> is finitely representable in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{p}.}</annotation>

</semantics>

</math><img src="./5de985eb969ec4307cee491238aa9aa19fc54c7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.676ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}.}" loading="lazy">

A Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is super-reflexive if all Banach spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> finitely representable in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> are reflexive, or, in other words, if no non-reflexive space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is finitely representable in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"> The notion of <a href="Ultraproduct" title="Ultraproduct">ultraproduct</a> of a family of Banach spaces<a href="#cite_note-15">[14]</a>

allows for a concise definition: the Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is super-reflexive when its ultrapowers are reflexive.

James proved that a space is super-reflexive if and only if its dual is super-reflexive.<a href="#cite_note-SRBS-14">[13]</a>

<div class="mw-heading mw-heading3"><h3 id="Finite_trees_in_Banach_spaces">Finite trees in Banach spaces</h3></div>

One of James' characterizations of super-reflexivity uses the growth of separated trees.<a href="#cite_note-Tree-16">[15]</a>

The description of a vectorial binary tree begins with a <a href="Rooted_binary_tree" class="mw-redirect" title="Rooted binary tree">rooted binary tree</a> labeled by vectors: a tree of <a href="Tree_(graph_theory)#Rooted_tree" title="Tree (graph theory)">height</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> in a Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a family of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n+1}-1}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 2^{n+1}-1}</annotation>

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</math><img src="./62451b1baffd4fa153d1a7465d21ea608205076d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.484ex; height:2.843ex;" alt="{\displaystyle 2^{n+1}-1}" loading="lazy"> vectors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

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</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> that can be organized in successive levels, starting with level 0 that consists of a single vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\varnothing },}">

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{\varnothing },}</annotation>

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</math><img src="./2b8753c0f5ad9d74ae479e3989977e03854f5802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.487ex; height:2.009ex;" alt="{\displaystyle x_{\varnothing },}" loading="lazy"> the <a href="Tree_(graph_theory)#Definitions" title="Tree (graph theory)">root</a> of the tree, followed, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,\ldots ,n,}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=1,\ldots ,n,}</annotation>

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</math><img src="./9a5700ec990a1e38bf455717c5570bed1445bcb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.692ex; height:2.509ex;" alt="{\displaystyle k=1,\ldots ,n,}" loading="lazy"> by a family of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s^{k}}</annotation>

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</math><img src="./640a2ea5ea628ce0bc033ac60e178a516c0cd240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.179ex; height:2.676ex;" alt="{\displaystyle s^{k}}" loading="lazy">2 vectors forming level <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k:}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle k:}</annotation>

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</math><img src="./fe720fa72a738af007689bd9f17a8aae79b890a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.503ex; height:2.176ex;" alt="{\displaystyle k:}" loading="lazy">

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \left\{x_{\varepsilon _{1},\ldots ,\varepsilon _{k}}\right\},\quad \varepsilon _{j}=\pm 1,\quad j=1,\ldots ,k,}</annotation>

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</math>

that are the <a href="Tree_(graph_theory)#Definitions" title="Tree (graph theory)">children</a> of vertices of level <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k-1.}">

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<annotation encoding="application/x-tex">{\displaystyle k-1.}</annotation>

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</math><img src="./c51f394a0e0100cf4d267b3d9858960b408cfc85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.861ex; height:2.343ex;" alt="{\displaystyle k-1.}" loading="lazy"> In addition to the <a href="Tree_(graph_theory)" title="Tree (graph theory)">tree structure</a>, it is required here that each vector that is an <a href="Tree_(graph_theory)#Definitions" title="Tree (graph theory)">internal vertex</a> of the tree be the midpoint between its two children:

<msub>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle x_{\emptyset }={\frac {x_{1}+x_{-1}}{2}},\quad x_{\varepsilon _{1},\ldots ,\varepsilon _{k}}={\frac {x_{\varepsilon _{1},\ldots ,\varepsilon _{k},1}+x_{\varepsilon _{1},\ldots ,\varepsilon _{k},-1}}{2}},\quad 1\leq k<n.}</annotation>

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</math>

Given a positive real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t,}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle t,}</annotation>

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</math><img src="./4ea3ad87830a1055c7b85c04cf940cfd3b847ae6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.486ex; height:2.343ex;" alt="{\displaystyle t,}" loading="lazy"> the tree is said to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">-separated if for every internal vertex, the two children are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">-separated in the given space norm:

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \left\|x_{1}-x_{-1}\right\|\geq t,\quad \left\|x_{\varepsilon _{1},\ldots ,\varepsilon _{k},1}-x_{\varepsilon _{1},\ldots ,\varepsilon _{k},-1}\right\|\geq t,\quad 1\leq k<n.}</annotation>

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</math>

<blockquote>Theorem.<a href="#cite_note-Tree-16">[15]</a>

The Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is super-reflexive if and only if for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in (0,2\pi ],}">

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<annotation encoding="application/x-tex">{\displaystyle t\in (0,2\pi ],}</annotation>

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</math><img src="./f7c703ee7b6ce6c22721cead5ea1c196f60d4c84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.569ex; height:2.843ex;" alt="{\displaystyle t\in (0,2\pi ],}" loading="lazy"> there is a number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n(t)}</annotation>

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</math><img src="./c405f3e8a5e549ba2d9cfe4dc76faded806c2b53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.044ex; height:2.843ex;" alt="{\displaystyle n(t)}" loading="lazy"> such that every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">-separated tree contained in the unit ball of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> has height less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(t).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n(t).}</annotation>

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</math><img src="./f76665c179a91878b4a4d579f553fa84adb8647c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.69ex; height:2.843ex;" alt="{\displaystyle n(t).}" loading="lazy"></blockquote>

<a href="Uniformly_convex_space" title="Uniformly convex space">Uniformly convex spaces</a> are super-reflexive.<a href="#cite_note-Tree-16">[15]</a>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> be uniformly convex, with <a href="Modulus_and_characteristic_of_convexity" title="Modulus and characteristic of convexity">modulus of convexity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{X}}</annotation>

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</math><img src="./dc60167425d241916b486aa52fd5303529661a4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.665ex; height:2.676ex;" alt="{\displaystyle \delta _{X}}" loading="lazy"> and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> be a real number in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,2].}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,2].}</annotation>

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</math><img src="./338e1fd062fe509b687b362c790045589c60d8e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.557ex; height:2.843ex;" alt="{\displaystyle (0,2].}" loading="lazy"> By the <a href="Modulus_and_characteristic_of_convexity#Definitions" title="Modulus and characteristic of convexity">properties</a> of the modulus of convexity, a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">-separated tree of height <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n,}</annotation>

</semantics>

</math><img src="./397bfafc701afdf14c2743278a097f6f2957eabb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.042ex; height:2.009ex;" alt="{\displaystyle n,}" loading="lazy"> contained in the unit ball, must have all points of level <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>

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</math><img src="./fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy"> contained in the ball of radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\delta _{X}(t)<1.}">

<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle 1-\delta _{X}(t)<1.}</annotation>

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</math><img src="./9d18f3e7d8085345c2a0ed8df8584a416217ab22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.224ex; height:2.843ex;" alt="{\displaystyle 1-\delta _{X}(t)<1.}" loading="lazy"> By induction, it follows that all points of level <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k}">

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<annotation encoding="application/x-tex">{\displaystyle n-k}</annotation>

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</math><img src="./b98e1d6a69bccd09a4b9b69bdf03a08c1706c8c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.446ex; height:2.343ex;" alt="{\displaystyle n-k}" loading="lazy"> are contained in the ball of radius

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \left(1-\delta _{X}(t)\right)^{j},\ j=1,\ldots ,n.}</annotation>

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If the height <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> was so large that

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \left(1-\delta _{X}(t)\right)^{n-1}<t/2,}</annotation>

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then the two points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},x_{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{1},x_{-1}}</annotation>

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</math><img src="./2e36dc0a1020380691ba41f7f91a6b2697c20f26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.08ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{-1}}" loading="lazy"> of the first level could not be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">-separated, contrary to the assumption. This gives the required bound <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(t),}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle n(t),}</annotation>

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</math><img src="./8999931d2e0a85c43126ae028f77f68a9ad9c772.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.69ex; height:2.843ex;" alt="{\displaystyle n(t),}" loading="lazy"> function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{X}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle \delta _{X}(t)}</annotation>

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</math><img src="./568d6cb4742068c8481d4ba82f7290c55bfa9500.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.314ex; height:2.843ex;" alt="{\displaystyle \delta _{X}(t)}" loading="lazy"> only.

Using the tree-characterization, <a href="Per_Enflo" title="Per Enflo">Enflo</a> proved<a href="#cite_note-17">[16]</a>

that super-reflexive Banach spaces admit an equivalent uniformly convex norm. Trees in a Banach space are a special instance of vector-valued <a href="Martingale_(probability_theory)" title="Martingale (probability theory)">martingales</a>. Adding techniques from scalar martingale theory, <a href="Gilles_Pisier" title="Gilles Pisier">Pisier</a> improved Enflo's result by showing<a href="#cite_note-18">[17]</a> that a super-reflexive space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle c>0}</annotation>

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</math><img src="./2ba126f626d61752f62eaacaf11761a54de4dc84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>0}" loading="lazy"> and some real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\geq 2,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\geq 2,}</annotation>

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</math><img src="./d6d8555c8f11bacabb1c9f3c233352ffbe9b56ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.977ex; height:2.509ex;" alt="{\displaystyle q\geq 2,}" loading="lazy">

<msub>

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<msup>

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</msup>

<mtext> whenever </mtext>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{X}(t)\geq c\,t^{q},\quad {\text{ whenever }}t\in [0,2].}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Reflexive_locally_convex_spaces">Reflexive locally convex spaces</h2></div>

The notion of reflexive Banach space can be generalized to <a href="Topological_vector_space" title="Topological vector space">topological vector spaces</a> in the following way.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> be a topological vector space over a number field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} }</annotation>

</semantics>

</math><img src="./573f72afae7df709959ab1a58cd643743466a187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {F} }" loading="lazy"> (of <a href="Real_number" title="Real number">real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> or <a href="Complex_number" title="Complex number">complex numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">). Consider its <a href="Strong_topology_(polar_topology)" class="mw-redirect" title="Strong topology (polar topology)">strong dual space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{b}^{\prime },}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{b}^{\prime },}</annotation>

</semantics>

</math><img src="./061cde7512c2ab6921f03a58695bf032637ab153.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.509ex; height:2.843ex;" alt="{\displaystyle X_{b}^{\prime },}" loading="lazy"> which consists of all <a href="Continuous_function" title="Continuous function">continuous</a> <a href="Linear_functional" class="mw-redirect" title="Linear functional">linear functionals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to \mathbb {F} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to \mathbb {F} }</annotation>

</semantics>

</math><img src="./3a8a1718f96d2610a6fe230d7533cf2f4437fa27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.23ex; height:2.509ex;" alt="{\displaystyle f:X\to \mathbb {F} }" loading="lazy"> and is equipped with the <a href="Strong_topology_(polar_topology)" class="mw-redirect" title="Strong topology (polar topology)">strong topology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\left(X^{\prime },X\right),}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b\left(X^{\prime },X\right),}</annotation>

</semantics>

</math><img src="./c11b6d858138b93ff51c36fbe049e163093d4be4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.923ex; height:3.009ex;" alt="{\displaystyle b\left(X^{\prime },X\right),}" loading="lazy"> that is,, the topology of uniform convergence on bounded subsets in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"> The space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{b}^{\prime }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{b}^{\prime }}</annotation>

</semantics>

</math><img src="./1852874a49781f1d80418a3946541499736a9611.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.862ex; height:2.843ex;" alt="{\displaystyle X_{b}^{\prime }}" loading="lazy"> is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime },}">

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime },}</annotation>

</semantics>

</math><img src="./5afa362e928e34f24b1e9d6f8024443fcdb6b925.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.576ex; height:3.676ex;" alt="{\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime },}" loading="lazy"> which is called the strong bidual space for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"> It consists of all continuous linear functionals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:X_{b}^{\prime }\to \mathbb {F} }">

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h:X_{b}^{\prime }\to \mathbb {F} }</annotation>

</semantics>

</math><img src="./0e80dffaff759dbc127263d78a82e76f7bbd81f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.172ex; height:2.843ex;" alt="{\displaystyle h:X_{b}^{\prime }\to \mathbb {F} }" loading="lazy"> and is equipped with the strong topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\left(\left(X_{b}^{\prime }\right)^{\prime },X_{b}^{\prime }\right).}">

<mrow>

<mrow>

<msup>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b\left(\left(X_{b}^{\prime }\right)^{\prime },X_{b}^{\prime }\right).}</annotation>

</semantics>

</math><img src="./d528186ae487c30988eb9b6314bd9e0ce77d34de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.766ex; height:4.843ex;" alt="{\displaystyle b\left(\left(X_{b}^{\prime }\right)^{\prime },X_{b}^{\prime }\right).}" loading="lazy"> Each vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> generates a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(x):X_{b}^{\prime }\to \mathbb {F} }">

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(x):X_{b}^{\prime }\to \mathbb {F} }</annotation>

</semantics>

</math><img src="./7843c4c284f0f3d77fb72c1859bd6520d9bc1130.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.444ex; height:3.009ex;" alt="{\displaystyle J(x):X_{b}^{\prime }\to \mathbb {F} }" loading="lazy"> by the following formula:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(x)(f)=f(x),\qquad f\in X^{\prime }.}</annotation>

</semantics>

</math>

This is a continuous linear functional on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{b}^{\prime },}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{b}^{\prime },}</annotation>

</semantics>

</math><img src="./061cde7512c2ab6921f03a58695bf032637ab153.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.509ex; height:2.843ex;" alt="{\displaystyle X_{b}^{\prime },}" loading="lazy"> that is,, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(x)\in \left(X_{b}^{\prime }\right)_{b}^{\prime }.}">

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(x)\in \left(X_{b}^{\prime }\right)_{b}^{\prime }.}</annotation>

</semantics>

</math><img src="./1e9e391b7414caaab69340e4a0280f2b5c6e7378.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.027ex; height:3.676ex;" alt="{\displaystyle J(x)\in \left(X_{b}^{\prime }\right)_{b}^{\prime }.}" loading="lazy"> This induces a map called the evaluation map:

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }.}</annotation>

</semantics>

</math>

This map is linear. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is locally convex, from the <a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach theorem</a> it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

</semantics>

</math><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> is injective and open (that is, for each neighbourhood of zero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> there is a neighbourhood of zero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime }}">

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime }}</annotation>

</semantics>

</math><img src="./5e0c8a7a5f7bc52ee00ea3efd2400f9697677880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.929ex; height:3.676ex;" alt="{\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime }}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(U)\supseteq V\cap J(X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(U)\supseteq V\cap J(X)}</annotation>

</semantics>

</math><img src="./0ea2f35a1a4e04bca25a7a22649ccb29b0867d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.792ex; height:2.843ex;" alt="{\displaystyle J(U)\supseteq V\cap J(X)}" loading="lazy">). But it can be non-surjective and/or discontinuous.

A locally convex space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is called

<ul><li>semi-reflexive if the evaluation map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}">

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}</annotation>

</semantics>

</math><img src="./e94461566a5025b83aa5f974704f7ee5051ff62d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.932ex; height:3.676ex;" alt="{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}" loading="lazy"> is surjective (hence bijective),</li>

<li>reflexive if the evaluation map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}">

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}</annotation>

</semantics>

</math><img src="./e94461566a5025b83aa5f974704f7ee5051ff62d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.932ex; height:3.676ex;" alt="{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}" loading="lazy"> is surjective and continuous (in this case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

</semantics>

</math><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> is an isomorphism of topological vector spaces<a href="#cite_note-isomorphism-19">[18]</a>).</li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}

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</style><div class="math_theorem" style="">

Theorem<a href="#cite_note-FOOTNOTEEdwards19658.4.2-20">[19]</a>—A locally convex Hausdorff space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is semi-reflexive if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (X,X^{*})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (X,X^{*})}</annotation>

</semantics>

</math><img src="./035ca0be45d6b5bc555bcfb75cf994fea587a780.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle \sigma (X,X^{*})}" loading="lazy">-topology has the Heine–Borel property (i.e. weakly closed and bounded subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> are weakly compact).

</div>

Theorem<a href="#cite_note-FOOTNOTESchaefer19665.6,_5.5-21">[20]</a><a href="#cite_note-FOOTNOTEEdwards19658.4.5-22">[21]</a>—A locally convex space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is reflexive if and only if it is semi-reflexive and <a href="Barrelled_space" title="Barrelled space">barreled</a>.

</div>

Theorem<a href="#cite_note-FOOTNOTEEdwards19658.4.3-23">[22]</a>—The strong dual of a semireflexive space is barrelled.

</div>

Theorem<a href="#cite_note-FOOTNOTENariciBeckenstein2011488–491-24">[23]</a>—If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a Hausdorff locally convex space then the canonical injection from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> into its bidual is a topological embedding if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is <a href="Infrabarreled_space" class="mw-redirect" title="Infrabarreled space">infrabarreled</a>.

</div>

<div class="mw-heading mw-heading3"><h3 id="Semireflexive_spaces">Semireflexive spaces</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Semi-reflexive_space" title="Semi-reflexive space">Semi-reflexive space</a></div>

<div class="mw-heading mw-heading4"><h4 id="Characterizations">Characterizations</h4></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a Hausdorff locally convex space then the following are equivalent:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is semireflexive;</li>

<li>The weak topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> had the Heine-Borel property (that is, for the weak topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \left(X,X^{\prime }\right),}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma \left(X,X^{\prime }\right),}</annotation>

</semantics>

</math><img src="./4fd1f4602e32d798e97e959efe457ff39fed3cb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.256ex; height:3.009ex;" alt="{\displaystyle \sigma \left(X,X^{\prime }\right),}" loading="lazy"> every closed and bounded subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\sigma }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{\sigma }}</annotation>

</semantics>

</math><img src="./f8852c163826f758e5f9b8b9d896d71a07ef5ad9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.097ex; height:2.509ex;" alt="{\displaystyle X_{\sigma }}" loading="lazy"> is weakly compact).<a href="#cite_note-FOOTNOTETrèves2006372–374-1">[1]</a></li>

<li>If linear form on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime }}</annotation>

</semantics>

</math><img src="./d7a3a5819cc45f097de14b3ac5a8bedd902bc66d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X^{\prime }}" loading="lazy"> that continuous when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime }}</annotation>

</semantics>

</math><img src="./d7a3a5819cc45f097de14b3ac5a8bedd902bc66d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X^{\prime }}" loading="lazy"> has the strong dual topology, then it is continuous when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime }}</annotation>

</semantics>

</math><img src="./d7a3a5819cc45f097de14b3ac5a8bedd902bc66d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X^{\prime }}" loading="lazy"> has the weak topology;<a href="#cite_note-FOOTNOTESchaeferWolff1999144-25">[24]</a></li>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{\tau }^{\prime }}</annotation>

</semantics>

</math><img src="./bc2a2ea8ac9365b0f2625f64209dedd99773b12b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.006ex; height:2.509ex;" alt="{\displaystyle X_{\tau }^{\prime }}" loading="lazy"> is barreled;<a href="#cite_note-FOOTNOTESchaeferWolff1999144-25">[24]</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> with the weak topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \left(X,X^{\prime }\right)}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma \left(X,X^{\prime }\right)}</annotation>

</semantics>

</math><img src="./21d9b3f88a4fcf1c70ebb38c2c80a3ef665f3e9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.222ex; height:3.009ex;" alt="{\displaystyle \sigma \left(X,X^{\prime }\right)}" loading="lazy"> is <a href="Quasi-complete" class="mw-redirect" title="Quasi-complete">quasi-complete</a>.<a href="#cite_note-FOOTNOTESchaeferWolff1999144-25">[24]</a></li></ol>

<div class="mw-heading mw-heading3"><h3 id="Characterizations_of_reflexive_spaces">Characterizations of reflexive spaces</h3></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a Hausdorff locally convex space then the following are equivalent:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is reflexive;</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is <a href="Semireflexive_space" class="mw-redirect" title="Semireflexive space">semireflexive</a> and <a href="Infrabarreled_space" class="mw-redirect" title="Infrabarreled space">infrabarreled</a>;<a href="#cite_note-FOOTNOTENariciBeckenstein2011488–491-24">[23]</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is <a href="Semireflexive_space" class="mw-redirect" title="Semireflexive space">semireflexive</a> and <a href="Barreled_space" class="mw-redirect" title="Barreled space">barreled</a>;</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is <a href="Barreled_space" class="mw-redirect" title="Barreled space">barreled</a> and the weak topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> had the Heine-Borel property (that is, for the weak topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \left(X,X^{\prime }\right),}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma \left(X,X^{\prime }\right),}</annotation>

</semantics>

</math><img src="./4fd1f4602e32d798e97e959efe457ff39fed3cb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.256ex; height:3.009ex;" alt="{\displaystyle \sigma \left(X,X^{\prime }\right),}" loading="lazy"> every closed and bounded subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\sigma }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{\sigma }}</annotation>

</semantics>

</math><img src="./f8852c163826f758e5f9b8b9d896d71a07ef5ad9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.097ex; height:2.509ex;" alt="{\displaystyle X_{\sigma }}" loading="lazy"> is weakly compact).<a href="#cite_note-FOOTNOTETrèves2006372–374-1">[1]</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is <a href="Semireflexive_space" class="mw-redirect" title="Semireflexive space">semireflexive</a> and <a href="Quasibarrelled_space" class="mw-redirect" title="Quasibarrelled space">quasibarrelled</a>.<a href="#cite_note-FOOTNOTEKhaleelulla198232–63-26">[25]</a></li></ol>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a normed space then the following are equivalent:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is reflexive;</li>

<li>The closed unit ball is compact when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> has the weak topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \left(X,X^{\prime }\right).}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma \left(X,X^{\prime }\right).}</annotation>

</semantics>

</math><img src="./061b357758b2bd99c641c77638169fae8fa63e83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.256ex; height:3.009ex;" alt="{\displaystyle \sigma \left(X,X^{\prime }\right).}" loading="lazy"><a href="#cite_note-FOOTNOTETrèves2006376-27">[26]</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a Banach space and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{b}^{\prime }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{b}^{\prime }}</annotation>

</semantics>

</math><img src="./1852874a49781f1d80418a3946541499736a9611.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.862ex; height:2.843ex;" alt="{\displaystyle X_{b}^{\prime }}" loading="lazy"> is reflexive.<a href="#cite_note-FOOTNOTETrèves2006377-28">[27]</a></li>

<li>Every sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(C_{n}\right)_{n=1}^{\infty },}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(C_{n}\right)_{n=1}^{\infty },}</annotation>

</semantics>

</math><img src="./6a131e2967cb17522e4dafa6301c17cebb8e655c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.655ex; height:3.176ex;" alt="{\displaystyle \left(C_{n}\right)_{n=1}^{\infty },}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n+1}\subseteq C_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{n+1}\subseteq C_{n}}</annotation>

</semantics>

</math><img src="./e87795e0a36633f946d2a6911f24fb80e13a0435.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.96ex; height:2.509ex;" alt="{\displaystyle C_{n+1}\subseteq C_{n}}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> of nonempty closed bounded convex subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> has nonempty intersection.<a href="#cite_note-FOOTNOTEBernardes2012-29">[28]</a></li></ol>

Theorem<a href="#cite_note-FOOTNOTENariciBeckenstein2011212-30">[29]</a>—A real Banach space is reflexive if and only if every pair of non-empty disjoint closed convex subsets, one of which is bounded, can be <a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">strictly separated by a hyperplane</a>.

</div>

<a href="James's_theorem" title="James's theorem">James' theorem</a>—A <a href="Banach_space" title="Banach space">Banach space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> is reflexive if and only if every <a href="Continuous_function" title="Continuous function">continuous</a> <a href="Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> attains its <a href="Supremum" class="mw-redirect" title="Supremum">supremum</a> on the closed <a href="Unit_ball" class="mw-redirect" title="Unit ball">unit ball</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B.}</annotation>

</semantics>

</math><img src="./0eccf5bca7cdc1fa4439af2d31831db6bde00473.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle B.}" loading="lazy">

</div>

<div class="mw-heading mw-heading3"><h3 id="Sufficient_conditions">Sufficient conditions</h3></div>

<dl><dt>Normed spaces</dt></dl>

A normed space that is semireflexive is a reflexive Banach space.<a href="#cite_note-FOOTNOTETrèves2006375-31">[30]</a>

A closed vector subspace of a reflexive Banach space is reflexive.<a href="#cite_note-FOOTNOTENariciBeckenstein2011488–491-24">[23]</a>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> be a Banach space and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> a closed vector subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"> If two of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,M,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,M,}</annotation>

</semantics>

</math><img src="./810e869bbf032bd0d064ad7735d084525b6fb6a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.103ex; height:2.509ex;" alt="{\displaystyle X,M,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/M}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X/M}</annotation>

</semantics>

</math><img src="./3aaa4313768efcc32d3d5a1de282d7bbe08baec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.585ex; height:2.843ex;" alt="{\displaystyle X/M}" loading="lazy"> are reflexive then they all are.<a href="#cite_note-FOOTNOTENariciBeckenstein2011488–491-24">[23]</a> This is why reflexivity is referred to as a three-space property.<a href="#cite_note-FOOTNOTENariciBeckenstein2011488–491-24">[23]</a>

<dl><dt>Topological vector spaces</dt></dl>

If a <a href="Barreled_space" class="mw-redirect" title="Barreled space">barreled</a> locally convex Hausdorff space is semireflexive then it is reflexive.<a href="#cite_note-FOOTNOTETrèves2006372–374-1">[1]</a>

The strong dual of a reflexive space is reflexive.<a href="#cite_note-FOOTNOTESchaeferWolff1999145-32">[31]</a>Every <a href="Montel_space" title="Montel space">Montel space</a> is reflexive.<a href="#cite_note-FOOTNOTETrèves2006376-27">[26]</a> And the strong dual of a <a href="Montel_space" title="Montel space">Montel space</a> is a Montel space (and thus is reflexive).<a href="#cite_note-FOOTNOTETrèves2006376-27">[26]</a>

<div class="mw-heading mw-heading3"><h3 id="Properties_2">Properties</h3></div>

A locally convex Hausdorff reflexive space is <a href="Barrelled_space" title="Barrelled space">barrelled</a>.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a normed space then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I:X\to X^{\prime \prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I:X\to X^{\prime \prime }}</annotation>

</semantics>

</math><img src="./2e0fd06d805cef4e2e0aee10bcc437a5f545a9fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.837ex; height:2.509ex;" alt="{\displaystyle I:X\to X^{\prime \prime }}" loading="lazy"> is an isometry onto a closed subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime \prime }.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime \prime }.}</annotation>

</semantics>

</math><img src="./cfae32fc5d9dfc08280fa75d84ce313cfd603cb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.781ex; height:2.509ex;" alt="{\displaystyle X^{\prime \prime }.}" loading="lazy"><a href="#cite_note-FOOTNOTETrèves2006375-31">[30]</a> This isometry can be expressed by:

<mover>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mover>

</mrow>

</munder>

<mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|x\|=\sup _{\stackrel {x^{\prime }\in X^{\prime },}{\|x^{\prime }\|\leq 1}}\left|\left\langle x^{\prime },x\right\rangle \right|.}</annotation>

</semantics>

</math>

Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a normed space and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime \prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime \prime }}</annotation>

</semantics>

</math><img src="./9f905bf6c4a877c966241e9d7650d10e3978b3dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.134ex; height:2.509ex;" alt="{\displaystyle X^{\prime \prime }}" loading="lazy"> is its bidual equipped with the bidual norm. Then the unit ball of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I(\{x\in X:\|x\|\leq 1\})}</annotation>

</semantics>

</math><img src="./aa2717370e727b35c5235ed0437bad8d7efdda11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.309ex; height:2.843ex;" alt="{\displaystyle I(\{x\in X:\|x\|\leq 1\})}" loading="lazy">

is dense in the unit ball

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{x^{\prime \prime }\in X^{\prime \prime }:\left\|x^{\prime \prime }\right\|\leq 1\right\}}</annotation>

</semantics>

</math><img src="./e81d21000c580cd378547ae29324f8cb6b83123b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.756ex; height:3.009ex;" alt="{\displaystyle \left\{x^{\prime \prime }\in X^{\prime \prime }:\left\|x^{\prime \prime }\right\|\leq 1\right\}}" loading="lazy">

of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime \prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime \prime }}</annotation>

</semantics>

</math><img src="./9f905bf6c4a877c966241e9d7650d10e3978b3dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.134ex; height:2.509ex;" alt="{\displaystyle X^{\prime \prime }}" loading="lazy"> for the weak topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right).}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right).}</annotation>

</semantics>

</math><img src="./a1a9f795c37d22bb5aee7e7936006ffaadd3089a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.41ex; height:3.009ex;" alt="{\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right).}" loading="lazy"><a href="#cite_note-FOOTNOTETrèves2006375-31">[30]</a>

<div class="mw-heading mw-heading3"><h3 id="Examples_2">Examples</h3></div>

<ol>

<li> Every finite-dimensional Hausdorff <a href="Topological_vector_space" title="Topological vector space">topological vector space</a> is reflexive, because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

</semantics>

</math><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.</li>

<li> A normed space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is reflexive as a normed space if and only if it is reflexive as a locally convex space. This follows from the fact that for a normed space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> its dual normed space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime }}</annotation>

</semantics>

</math><img src="./d7a3a5819cc45f097de14b3ac5a8bedd902bc66d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X^{\prime }}" loading="lazy"> coincides as a topological vector space with the strong dual space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{b}^{\prime }.}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{b}^{\prime }.}</annotation>

</semantics>

</math><img src="./551d2a874fd662ca1053d59f3e1f4319d93a20e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.509ex; height:2.843ex;" alt="{\displaystyle X_{b}^{\prime }.}" loading="lazy"> As a corollary, the evaluation map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J:X\to X^{\prime \prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to X^{\prime \prime }}</annotation>

</semantics>

</math><img src="./c51ba0fa314c691e9410e8807b52f4a8dc98fc6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.137ex; height:2.509ex;" alt="{\displaystyle J:X\to X^{\prime \prime }}" loading="lazy"> coincides with the evaluation map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime },}">

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime },}</annotation>

</semantics>

</math><img src="./3b16848c011210599eea40a067c99d099290ba42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.579ex; height:3.676ex;" alt="{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime },}" loading="lazy"> and the following conditions become equivalent:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a reflexive normed space (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J:X\to X^{\prime \prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to X^{\prime \prime }}</annotation>

</semantics>

</math><img src="./c51ba0fa314c691e9410e8807b52f4a8dc98fc6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.137ex; height:2.509ex;" alt="{\displaystyle J:X\to X^{\prime \prime }}" loading="lazy"> is an isomorphism of normed spaces),</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a reflexive locally convex space (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}">

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}</annotation>

</semantics>

</math><img src="./e94461566a5025b83aa5f974704f7ee5051ff62d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.932ex; height:3.676ex;" alt="{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}" loading="lazy"> is an isomorphism of topological vector spaces<a href="#cite_note-isomorphism-19">[18]</a>),</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a semi-reflexive locally convex space (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}">

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}</annotation>

</semantics>

</math><img src="./e94461566a5025b83aa5f974704f7ee5051ff62d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.932ex; height:3.676ex;" alt="{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}" loading="lazy"> is surjective).</li>

</ol>

</li>

<li>A (somewhat artificial) example of a semi-reflexive space that is not reflexive is obtained as follows: let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> be an infinite dimensional reflexive Banach space, and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> be the topological vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(Y,\sigma \left(Y,Y^{\prime }\right)\right),}">

<mrow>

<mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(Y,\sigma \left(Y,Y^{\prime }\right)\right),}</annotation>

</semantics>

</math><img src="./2c509d32f2b3b08953f8b6ca833fef567810665c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.569ex; height:3.009ex;" alt="{\displaystyle \left(Y,\sigma \left(Y,Y^{\prime }\right)\right),}" loading="lazy"> that is, the vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> equipped with the weak topology. Then the continuous dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y^{\prime }}</annotation>

</semantics>

</math><img src="./dc6b95f963ad415c9e50171ecb1826cc4c094477.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.585ex; height:2.343ex;" alt="{\displaystyle Y^{\prime }}" loading="lazy"> are the same set of functionals, and bounded subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (that is, weakly bounded subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">) are norm-bounded, hence the Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y^{\prime }}</annotation>

</semantics>

</math><img src="./dc6b95f963ad415c9e50171ecb1826cc4c094477.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.585ex; height:2.343ex;" alt="{\displaystyle Y^{\prime }}" loading="lazy"> is the strong dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is reflexive, the continuous dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime }=Y^{\prime }}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime }=Y^{\prime }}</annotation>

</semantics>

</math><img src="./94a8529cf793526283e8a1e4225afddb1f8b380f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.365ex; height:2.509ex;" alt="{\displaystyle X^{\prime }=Y^{\prime }}" loading="lazy"> is equal to the image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(X)}</annotation>

</semantics>

</math><img src="./c1ec6aa47953554d0e553013321bdbbc3dab105a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.261ex; height:2.843ex;" alt="{\displaystyle J(X)}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> under the canonical embedding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J,}</annotation>

</semantics>

</math><img src="./5eea287db0bfbce6f45e1cf373b3d642f2993ef0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.118ex; height:2.509ex;" alt="{\displaystyle J,}" loading="lazy"> but the topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (the weak topology of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">) is not the strong topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \left(X,X^{\prime }\right),}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta \left(X,X^{\prime }\right),}</annotation>

</semantics>

</math><img src="./a195c1a6de58b719b3937510a091b282b3fa949a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.258ex; height:3.009ex;" alt="{\displaystyle \beta \left(X,X^{\prime }\right),}" loading="lazy"> that is equal to the norm topology of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y.}</annotation>

</semantics>

</math><img src="./0c668649af47a30006f93c9847d61fee8d9ffb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" loading="lazy"></li>

<li><a href="Montel_space" title="Montel space">Montel spaces</a> are reflexive locally convex topological vector spaces. In particular, the following functional spaces frequently used in functional analysis are reflexive locally convex spaces:<a href="#cite_note-33">[32]</a>

<ul><li>the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(M)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{\infty }(M)}</annotation>

</semantics>

</math><img src="./48b543485314a4426ee2ff956f206cfd802d68d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.925ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(M)}" loading="lazy"> of smooth functions on arbitrary (real) smooth manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M,}</annotation>

</semantics>

</math><img src="./b466e90209f39c0c2caad1b11445824b82c2f536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.089ex; height:2.509ex;" alt="{\displaystyle M,}" loading="lazy"> and its strong dual space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(C^{\infty }\right)^{\prime }(M)}">

<msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(C^{\infty }\right)^{\prime }(M)}</annotation>

</semantics>

</math><img src="./c2523ea6f9f8f68f831b97854d532fcc9b6c922d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.419ex; height:3.176ex;" alt="{\displaystyle \left(C^{\infty }\right)^{\prime }(M)}" loading="lazy"> of distributions with compact support on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M,}</annotation>

</semantics>

</math><img src="./b466e90209f39c0c2caad1b11445824b82c2f536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.089ex; height:2.509ex;" alt="{\displaystyle M,}" loading="lazy"></li>

<li>the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}(M)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}(M)}</annotation>

</semantics>

</math><img src="./e37f0e0dff00c74a5afda7ed4dc65d645db3b4bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.043ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}(M)}" loading="lazy"> of smooth functions with compact support on arbitrary (real) smooth manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M,}</annotation>

</semantics>

</math><img src="./b466e90209f39c0c2caad1b11445824b82c2f536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.089ex; height:2.509ex;" alt="{\displaystyle M,}" loading="lazy"> and its strong dual space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}^{\prime }(M)}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}^{\prime }(M)}</annotation>

</semantics>

</math><img src="./ed5be9df01257d3f1c7e88e1fc7606346cc48e8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.728ex; height:3.009ex;" alt="{\displaystyle {\mathcal {D}}^{\prime }(M)}" loading="lazy"> of distributions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M,}</annotation>

</semantics>

</math><img src="./b466e90209f39c0c2caad1b11445824b82c2f536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.089ex; height:2.509ex;" alt="{\displaystyle M,}" loading="lazy"></li>

<li>the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(M)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(M)}</annotation>

</semantics>

</math><img src="./84fff48af4c0694fcb3f73a58422215818e43dfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.101ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(M)}" loading="lazy"> of holomorphic functions on arbitrary complex manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M,}</annotation>

</semantics>

</math><img src="./b466e90209f39c0c2caad1b11445824b82c2f536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.089ex; height:2.509ex;" alt="{\displaystyle M,}" loading="lazy"> and its strong dual space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}^{\prime }(M)}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}^{\prime }(M)}</annotation>

</semantics>

</math><img src="./22cc57ff1d585b4b5e6740f2fa313a2d4bf643a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.786ex; height:3.009ex;" alt="{\displaystyle {\mathcal {O}}^{\prime }(M)}" loading="lazy"> of analytic functionals on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M,}</annotation>

</semantics>

</math><img src="./b466e90209f39c0c2caad1b11445824b82c2f536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.089ex; height:2.509ex;" alt="{\displaystyle M,}" loading="lazy"></li>

<li>the <a href="Schwartz_space" title="Schwartz space">Schwartz space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}">

</mrow>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}</annotation>

</semantics>

</math><img src="./c7b6abf90493483d3d936d32d9dc028d434f83c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.585ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n},}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n},}</annotation>

</semantics>

</math><img src="./d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n},}" loading="lazy"> and its strong dual space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}^{\prime }\left(\mathbb {R} ^{n}\right)}">

<msup>

</mrow>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}^{\prime }\left(\mathbb {R} ^{n}\right)}</annotation>

</semantics>

</math><img src="./e9280845e749767054cb15735931dcd2fa547ddc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.295ex; height:3.009ex;" alt="{\displaystyle {\mathcal {S}}^{\prime }\left(\mathbb {R} ^{n}\right)}" loading="lazy"> of tempered distributions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}.}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}.}</annotation>

</semantics>

</math><img src="./76ef548febfc9981762740107858be9e3a5576c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}.}" loading="lazy"></li></ul>

</li>

</ol>

<div class="mw-heading mw-heading3"><h3 id="Counter-examples">Counter-examples</h3></div>

<ul><li>There exists a non-reflexive locally convex TVS whose strong dual is reflexive.<a href="#cite_note-FOOTNOTESchaeferWolff1999190–202-34">[33]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Other_types_of_reflexivity">Other types of reflexivity</h2></div>

A stereotype space, or polar reflexive space, is defined as a <a href="Topological_vector_space" title="Topological vector space">topological vector space</a> (TVS) satisfying a similar condition of reflexivity, but with the topology of uniform convergence on <a href="Totally_bounded_set" class="mw-redirect" title="Totally bounded set">totally bounded</a> subsets (instead of <a href="Bounded_set" title="Bounded set">bounded</a> subsets) in the definition of dual space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime }.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime }.}</annotation>

</semantics>

</math><img src="./980210d5ccf78c678264901dc7b0ce8a53d827bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.328ex; height:2.509ex;" alt="{\displaystyle X^{\prime }.}" loading="lazy"> More precisely, a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is called polar reflexive<a href="#cite_note-35">[34]</a> or stereotype if the evaluation map into the second dual space

<msup>

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</msup>

<msup>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:X\to X^{\star \star },\quad J(x)(f)=f(x),\quad x\in X,\quad f\in X^{\star }}</annotation>

</semantics>

</math>

is an <a href="TVS-isomorphism" class="mw-redirect" title="TVS-isomorphism">isomorphism of topological vector spaces</a>.<a href="#cite_note-isomorphism-19">[18]</a> Here the stereotype dual space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\star }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\star }}</annotation>

</semantics>

</math><img src="./b7e61f747e46a210fc53ab24d3c136201e42339a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.343ex;" alt="{\displaystyle X^{\star }}" loading="lazy"> is defined as the space of continuous linear functionals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime }}</annotation>

</semantics>

</math><img src="./d7a3a5819cc45f097de14b3ac5a8bedd902bc66d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X^{\prime }}" loading="lazy"> endowed with the topology of uniform convergence on totally bounded sets in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (and the stereotype second dual space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\star \star }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\star \star }}</annotation>

</semantics>

</math><img src="./c40570ab611afb208ebd88b3ceb9077ac787eb82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.873ex; height:2.343ex;" alt="{\displaystyle X^{\star \star }}" loading="lazy"> is the space dual to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\star }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\star }}</annotation>

</semantics>

</math><img src="./b7e61f747e46a210fc53ab24d3c136201e42339a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.343ex;" alt="{\displaystyle X^{\star }}" loading="lazy"> in the same sense).

In contrast to the classical reflexive spaces the class Ste of stereotype spaces is very wide (it contains, in particular, all <a href="Fr%C3%A9chet_space" title="Fréchet space">Fréchet spaces</a> and thus, all <a href="Banach_space" title="Banach space">Banach spaces</a>), it forms a <a href="Closed_monoidal_category" title="Closed monoidal category">closed monoidal category</a>, and it admits standard operations (defined inside of Ste) of constructing new spaces, like taking closed subspaces, quotient spaces, projective and injective limits, the space of operators, tensor products, etc. The category Ste have applications in duality theory for non-commutative groups.

Similarly, one can replace the class of bounded (and totally bounded) subsets in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> in the definition of dual space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\prime },}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{\prime },}</annotation>

</semantics>

</math><img src="./90fef41e6ae66cd91995ce221cc99d754e3dd4ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.328ex; height:2.843ex;" alt="{\displaystyle X^{\prime },}" loading="lazy"> by other classes of subsets, for example, by the class of compact subsets in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> – the spaces defined by the corresponding reflexivity condition are called reflective,<a href="#cite_note-36">[35]</a><a href="#cite_note-37">[36]</a> and they form an even wider class than Ste, but it is not clear (2012), whether this class forms a category with properties similar to those of Ste.

<ul><li><a href="Grothendieck_space" title="Grothendieck space">Grothendieck space</a>

<ul><li>A generalization which has some of the properties of reflexive spaces and includes many spaces of practical importance is the concept of <a href="Grothendieck_space" title="Grothendieck space">Grothendieck space</a>.</li></ul></li>

<li><a href="Reflexive_operator_algebra" title="Reflexive operator algebra">Reflexive operator algebra</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3></div>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> The statement of <a href="Riesz's_lemma" title="Riesz's lemma">Riesz's lemma</a> involves only one real number, which is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> in the article on Riesz's lemma. The lemma always holds for all real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <1.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha <1.}</annotation>

</semantics>

</math><img src="./caf19239981cb195e55e70640cd392f1181831b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.395ex; height:2.176ex;" alt="{\displaystyle \alpha <1.}" loading="lazy"> But for a Banach space, the lemma holds for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \leq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \leq 1}</annotation>

</semantics>

</math><img src="./445cb004cd7fa3dcd89b8d4d5ceebb11bb5db8e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.749ex; height:2.343ex;" alt="{\displaystyle \alpha \leq 1}" loading="lazy"> if and only if the space is reflexive.

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

<li id="cite_note-FOOTNOTETrèves2006372–374-1">^ <a href="#cite_ref-FOOTNOTETrèves2006372–374_1-0">a</a> <a href="#cite_ref-FOOTNOTETrèves2006372–374_1-1">b</a> <a href="#cite_ref-FOOTNOTETrèves2006372–374_1-2">c</a> <a href="#cite_ref-FOOTNOTETrèves2006372–374_1-3">d</a> <a href="#cite_ref-FOOTNOTETrèves2006372–374_1-4">e</a> <a href="#CITEREFTrèves2006">Trèves 2006</a>, pp. 372–374.

</li>

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</style><cite id="CITEREFRobert_C._James1951" class="citation journal cs1"><a href="Robert_C._James" title="Robert C. James">Robert C. James</a> (1951). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063327">"A non-reflexive Banach space isometric with its second conjugate space"</a>. Proc. Natl. Acad. Sci. U.S.A. 37 (3): 174–177. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1951PNAS...37..174J">1951PNAS...37..174J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.37.3.174">10.1073/pnas.37.3.174</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063327">1063327</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16588998">16588998</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Proposition 1.11.8 in <a href="#CITEREFMegginson1998">Megginson (1998</a>, p. 99).

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFMegginson1998">Megginson (1998</a>, pp. 104–105).

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Corollary 1.11.17, p. 104 in <a href="#CITEREFMegginson1998">Megginson (1998)</a>.

</li>

<li id="cite_note-FOOTNOTEConway1985Theorem_V.4.2,_p.&nbsp;135-6"><a href="#cite_ref-FOOTNOTEConway1985Theorem_V.4.2,_p.&nbsp;135_6-0">^</a> <a href="#CITEREFConway1985">Conway 1985</a>, Theorem V.4.2, p. 135.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> Since weak compactness and weak sequential compactness coincide by the <a href="Eberlein%E2%80%93%C5%A0mulian_theorem" title="Eberlein–Šmulian theorem">Eberlein–Šmulian theorem</a>.

</li>

<li id="cite_note-FOOTNOTEDiestel19846-9"><a href="#cite_ref-FOOTNOTEDiestel19846_9-0">^</a> <a href="#CITEREFDiestel1984">Diestel 1984</a>, p. 6.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> Theorem 1.13.11 in <a href="#CITEREFMegginson1998">Megginson (1998</a>, p. 125).

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> Theorem 2.5.16 in <a href="#CITEREFMegginson1998">Megginson (1998</a>, p. 216).

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> Theorem 1.12.11 and Corollary 1.12.12 in <a href="#CITEREFMegginson1998">Megginson (1998</a>, pp. 112–113).

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> see this <a href="Banach_space#Characterizations_of_Hilbert_space_among_Banach_spaces" title="Banach space">characterization of Hilbert space among Banach spaces</a>

</li>

<li id="cite_note-SRBS-14">^ <a href="#cite_ref-SRBS_14-0">a</a> <a href="#cite_ref-SRBS_14-1">b</a> James, Robert C. (1972), "Super-reflexive Banach spaces", Can. J. Math. 24:896–904.

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> Dacunha-Castelle, Didier; Krivine, Jean-Louis (1972), "Applications des ultraproduits à l'étude des espaces et des algèbres de Banach" (in French), Studia Math. 41:315–334.

</li>

<li id="cite_note-Tree-16">^ <a href="#cite_ref-Tree_16-0">a</a> <a href="#cite_ref-Tree_16-1">b</a> <a href="#cite_ref-Tree_16-2">c</a> see <a href="#CITEREFJames1972">James (1972)</a>.

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFEnflo1972" class="citation journal cs1"><a href="Per_Enflo" title="Per Enflo">Enflo, Per</a> (1972). "Banach spaces which can be given an equivalent uniformly convex norm". <a href="Israel_Journal_of_Mathematics" title="Israel Journal of Mathematics">Israel Journal of Mathematics</a>. 13 (3–4): 281–288. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02762802">10.1007/BF02762802</a>.</cite>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFPisier1975" class="citation journal cs1"><a href="Gilles_Pisier" title="Gilles Pisier">Pisier, Gilles</a> (1975). "Martingales with values in uniformly convex spaces". <a href="Israel_Journal_of_Mathematics" title="Israel Journal of Mathematics">Israel Journal of Mathematics</a>. 20 (3–4): 326–350. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02760337">10.1007/BF02760337</a>.</cite>

</li>

<li id="cite_note-isomorphism-19">^ <a href="#cite_ref-isomorphism_19-0">a</a> <a href="#cite_ref-isomorphism_19-1">b</a> <a href="#cite_ref-isomorphism_19-2">c</a> An <a href="TVS-isomorphism" class="mw-redirect" title="TVS-isomorphism">isomorphism of topological vector spaces</a> is a <a href="Linear_map" title="Linear map">linear</a> and a <a href="Homeomorphism" title="Homeomorphism">homeomorphic</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :X\to Y.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi :X\to Y.}</annotation>

</semantics>

</math><img src="./f8313c98fe4c186c3308d5e8c667a603cff79b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.472ex; height:2.676ex;" alt="{\displaystyle \varphi :X\to Y.}" loading="lazy">

</li>

<li id="cite_note-FOOTNOTEEdwards19658.4.2-20"><a href="#cite_ref-FOOTNOTEEdwards19658.4.2_20-0">^</a> <a href="#CITEREFEdwards1965">Edwards 1965</a>, 8.4.2.

</li>

<li id="cite_note-FOOTNOTESchaefer19665.6,_5.5-21"><a href="#cite_ref-FOOTNOTESchaefer19665.6,_5.5_21-0">^</a> <a href="#CITEREFSchaefer1966">Schaefer 1966</a>, 5.6, 5.5.

</li>

<li id="cite_note-FOOTNOTEEdwards19658.4.5-22"><a href="#cite_ref-FOOTNOTEEdwards19658.4.5_22-0">^</a> <a href="#CITEREFEdwards1965">Edwards 1965</a>, 8.4.5.

</li>

<li id="cite_note-FOOTNOTEEdwards19658.4.3-23"><a href="#cite_ref-FOOTNOTEEdwards19658.4.3_23-0">^</a> <a href="#CITEREFEdwards1965">Edwards 1965</a>, 8.4.3.

</li>

<li id="cite_note-FOOTNOTENariciBeckenstein2011488–491-24">^ <a href="#cite_ref-FOOTNOTENariciBeckenstein2011488–491_24-0">a</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011488–491_24-1">b</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011488–491_24-2">c</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011488–491_24-3">d</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011488–491_24-4">e</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, pp. 488–491.

</li>

<li id="cite_note-FOOTNOTESchaeferWolff1999144-25">^ <a href="#cite_ref-FOOTNOTESchaeferWolff1999144_25-0">a</a> <a href="#cite_ref-FOOTNOTESchaeferWolff1999144_25-1">b</a> <a href="#cite_ref-FOOTNOTESchaeferWolff1999144_25-2">c</a> <a href="#CITEREFSchaeferWolff1999">Schaefer & Wolff 1999</a>, p. 144.

</li>

<li id="cite_note-FOOTNOTEKhaleelulla198232–63-26"><a href="#cite_ref-FOOTNOTEKhaleelulla198232–63_26-0">^</a> <a href="#CITEREFKhaleelulla1982">Khaleelulla 1982</a>, pp. 32–63.

</li>

<li id="cite_note-FOOTNOTETrèves2006376-27">^ <a href="#cite_ref-FOOTNOTETrèves2006376_27-0">a</a> <a href="#cite_ref-FOOTNOTETrèves2006376_27-1">b</a> <a href="#cite_ref-FOOTNOTETrèves2006376_27-2">c</a> <a href="#CITEREFTrèves2006">Trèves 2006</a>, p. 376.

</li>

<li id="cite_note-FOOTNOTETrèves2006377-28"><a href="#cite_ref-FOOTNOTETrèves2006377_28-0">^</a> <a href="#CITEREFTrèves2006">Trèves 2006</a>, p. 377.

</li>

<li id="cite_note-FOOTNOTEBernardes2012-29"><a href="#cite_ref-FOOTNOTEBernardes2012_29-0">^</a> <a href="#CITEREFBernardes2012">Bernardes 2012</a>.

</li>

<li id="cite_note-FOOTNOTENariciBeckenstein2011212-30"><a href="#cite_ref-FOOTNOTENariciBeckenstein2011212_30-0">^</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, pp. 212.

</li>

<li id="cite_note-FOOTNOTETrèves2006375-31">^ <a href="#cite_ref-FOOTNOTETrèves2006375_31-0">a</a> <a href="#cite_ref-FOOTNOTETrèves2006375_31-1">b</a> <a href="#cite_ref-FOOTNOTETrèves2006375_31-2">c</a> <a href="#CITEREFTrèves2006">Trèves 2006</a>, p. 375.

</li>

<li id="cite_note-FOOTNOTESchaeferWolff1999145-32"><a href="#cite_ref-FOOTNOTESchaeferWolff1999145_32-0">^</a> <a href="#CITEREFSchaeferWolff1999">Schaefer & Wolff 1999</a>, p. 145.

</li>

<li id="cite_note-33"><a href="#cite_ref-33">^</a> <a href="#CITEREFEdwards1965">Edwards 1965</a>, 8.4.7.

</li>

<li id="cite_note-FOOTNOTESchaeferWolff1999190–202-34"><a href="#cite_ref-FOOTNOTESchaeferWolff1999190–202_34-0">^</a> <a href="#CITEREFSchaeferWolff1999">Schaefer & Wolff 1999</a>, pp. 190–202.

</li>

<li id="cite_note-35"><a href="#cite_ref-35">^</a> <cite id="CITEREFKöthe1983" class="citation book cs1">Köthe, Gottfried (1983). <a rel="nofollow" class="external text" href="https://www.springer.com/gp/book/9783642649905#aboutBook">Topological Vector Spaces I</a>. Springer Grundlehren der mathematischen Wissenschaften. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-64988-2</bdi>.</cite>

</li>

<li id="cite_note-36"><a href="#cite_ref-36">^</a> <cite id="CITEREFGaribay_BonalesTrigos-Arrieta,_F._J.Vera_Mendoza,_R.2002" class="citation journal cs1">Garibay Bonales, F.; Trigos-Arrieta, F. J.; Vera Mendoza, R. (2002). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0166-8641%2801%2900111-0">"A characterization of Pontryagin-van Kampen duality for locally convex spaces"</a>. Topology and Its Applications. 121 (1–2): 75–89. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0166-8641%2801%2900111-0">10.1016/s0166-8641(01)00111-0</a>.</cite>

</li>

<li id="cite_note-37"><a href="#cite_ref-37">^</a> <cite id="CITEREFAkbarovShavgulidze,_E._T.2003" class="citation journal cs1">Akbarov, S. S.; Shavgulidze, E. T. (2003). "On two classes of spaces reflexive in the sense of Pontryagin". Mat. Sbornik. 194 (10): 3–26.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="General_references">General references</h3></div>

<ul><li><cite id="CITEREFBernardes2012" class="citation cs2">Bernardes, Nilson C. Jr. (2012), On nested sequences of convex sets in Banach spaces, vol. 389, Journal of Mathematical Analysis and Applications, pp. 558–561</cite> .</li>

<li><cite id="CITEREFConway1985" class="citation book cs1"><a href="John_B._Conway" title="John B. Conway">Conway, John B.</a> (1985). A Course in Functional Analysis. Springer.</cite></li>

<li><cite id="CITEREFDiestel1984" class="citation book cs1">Diestel, Joe (1984). Sequences and series in Banach spaces. New York: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-90859-5</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/9556781">9556781</a>.</cite></li>

<li><cite id="CITEREFEdwards1965" class="citation book cs1">Edwards, R. E. (1965). Functional analysis. Theory and applications. New York: Holt, Rinehart and Winston. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0030505356</bdi>.</cite></li>

<li><cite id="CITEREFJames1972" class="citation cs2">James, Robert C. (1972), Some self-dual properties of normed linear spaces. Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967), Ann. of Math. Studies, vol. 69, Princeton, NJ: Princeton Univ. Press, pp. 159–175</cite>.</li>

<li><cite id="CITEREFKhaleelulla1982" class="citation book cs1">Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. <a href="Lecture_Notes_in_Mathematics" title="Lecture Notes in Mathematics">Lecture Notes in Mathematics</a>. Vol. 936. Berlin, Heidelberg, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-11565-6</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/8588370">8588370</a>.</cite></li>

<li><cite id="CITEREFKolmogorovFomin1957" class="citation book cs1">Kolmogorov, A. N.; Fomin, S. V. (1957). Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces. Rochester: Graylock Press.</cite></li>

<li><cite id="CITEREFMegginson1998" class="citation cs2"><a href="Robert_Megginson" title="Robert Megginson">Megginson, Robert E.</a> (1998), An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, New York: Springer-Verlag, pp. xx+596, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98431-3</bdi></cite></li>

<li><cite id="CITEREFNariciBeckenstein2011" class="citation book cs1">Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1584888666</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/144216834">144216834</a>.</cite></li>

<li><cite id="CITEREFRudin1991" class="citation book cs1"><a href="Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi">Functional Analysis</a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="McGraw-Hill_Science/Engineering/Math" class="mw-redirect" title="McGraw-Hill Science/Engineering/Math">McGraw-Hill Science/Engineering/Math</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-054236-5</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/21163277">21163277</a>.</cite></li>

<li><cite id="CITEREFSchaefer1966" class="citation book cs1"><a href="Helmut_H._Schaefer" title="Helmut H. Schaefer">Schaefer, Helmut H.</a> (1966). Topological vector spaces. New York: The Macmillan Company.</cite></li>

<li><cite id="CITEREFSchaeferWolff1999" class="citation book cs1"><a href="Helmut_H._Schaefer" title="Helmut H. Schaefer">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). Topological Vector Spaces. <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4612-7155-0</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/840278135">840278135</a>.</cite></li>

<li><cite id="CITEREFTrèves2006" class="citation book cs1"><a href="Fran%C3%A7ois_Tr%C3%A8ves" title="François Trèves">Trèves, François</a> (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-45352-1</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/853623322">853623322</a>.</cite></li></ul>

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</style><div id="Banach_space_topics288" style="font-size:114%;margin:0 4em"><a href="Banach_space" title="Banach space">Banach space</a> topics</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of Banach spaces</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Asplund_space" title="Asplund space">Asplund</a></li>

<li><a href="Banach_space" title="Banach space">Banach</a>

<li><a href="Banach_lattice" title="Banach lattice">Banach lattice</a></li>

<li><a href="Grothendieck_space" title="Grothendieck space">Grothendieck </a></li>

<li><a href="Hilbert_space" title="Hilbert space">Hilbert</a>

<ul><li><a href="Inner_product_space" title="Inner product space">Inner product space</a></li>

<li><a href="Polarization_identity" title="Polarization identity">Polarization identity</a></li></ul></li>

<li>(<a href="Polynomially_reflexive_space" title="Polynomially reflexive space">Polynomially</a>) </li>

<li><a href="Riesz_space" title="Riesz space">Riesz</a></li>

<li><a href="L-semi-inner_product" title="L-semi-inner product">L-semi-inner product</a></li>

<li><a href="Strictly_convex_space" title="Strictly convex space">Strictly</a></li>

<li><a href="Uniformly_convex_space" title="Uniformly convex space">Uniformly</a>) convex</li>

<li><a href="Uniformly_smooth_space" title="Uniformly smooth space">Uniformly smooth</a></li>

<li>(<a href="Injective_tensor_product" title="Injective tensor product">Injective</a></li>

<li><a href="Projective_tensor_product" title="Projective tensor product">Projective</a>) <a href="Topological_tensor_product" title="Topological tensor product">Tensor product</a> (<a href="Tensor_product_of_Hilbert_spaces" title="Tensor product of Hilbert spaces">of Hilbert spaces</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Banach spaces are:</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Barrelled_space" title="Barrelled space">Barrelled</a></li>

<li><a href="Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li>

<li><a href="F-space" title="F-space">F-space</a></li>

<li><a href="Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a>

<li><a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a>

<ul><li><a href="Locally_convex_topological_vector_space#Definition_via_seminorms" title="Locally convex topological vector space">Seminorms</a>/<a href="Minkowski_functional" title="Minkowski functional">Minkowski functionals</a></li></ul></li>

<li><a href="Mackey_space" title="Mackey space">Mackey</a></li>

<li><a href="Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizable</a></li>

<li><a href="Normed_space" class="mw-redirect" title="Normed space">Normed</a>

<li><a href="Quasinorm" title="Quasinorm">Quasinormed</a></li>

<li><a href="Stereotype_space" class="mw-redirect" title="Stereotype space">Stereotype</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Function space Topologies</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach%E2%80%93Mazur_compactum" title="Banach–Mazur compactum">Banach–Mazur compactum</a></li>

<li><a href="Dual_space" title="Dual space">Dual space</a>

<li><a href="Operator_topologies" title="Operator topologies">Operator</a></li>

<li><a href="Ultraweak_topology" title="Ultraweak topology">Ultraweak</a></li>

<ul><li><a href="Weak_topology_(polar_topology)" class="mw-redirect" title="Weak topology (polar topology)">polar</a></li>

<li><a href="Weak_operator_topology" title="Weak operator topology">operator</a></li></ul></li>

<li><a href="Strong_topology" title="Strong topology">Strong</a>

<ul><li><a href="Strong_topology_(polar_topology)" class="mw-redirect" title="Strong topology (polar topology)">polar</a></li>

<li><a href="Strong_operator_topology" title="Strong operator topology">operator</a></li></ul></li>

<li><a href="Ultrastrong_topology" title="Ultrastrong topology">Ultrastrong</a></li>

<li><a href="Topology_of_uniform_convergence" class="mw-redirect" title="Topology of uniform convergence">Uniform convergence</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Linear_operator" class="mw-redirect" title="Linear operator">Linear operators</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hermitian_adjoint" title="Hermitian adjoint">Adjoint</a></li>

<li><a href="Bilinear_map" title="Bilinear map">Bilinear</a>

<li><a href="Bilinear_map" title="Bilinear map">operator</a></li>

<li><a href="Sesquilinear_form" title="Sesquilinear form">sesquilinear</a></li></ul></li>

<li>(<a href="Unbounded_operator" title="Unbounded operator">Un</a>)<a href="Bounded_operator" title="Bounded operator">Bounded</a></li>

<li><a href="Closed_linear_operator" title="Closed linear operator">Closed</a></li>

<li><a href="Compact_operator" title="Compact operator">Compact</a>

<ul><li><a href="Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">on Hilbert spaces</a></li></ul></li>

<li>(<a href="Discontinuous_linear_map" title="Discontinuous linear map">Dis</a>)<a href="Continuous_linear_operator" title="Continuous linear operator">Continuous</a></li>

<li><a href="Densely_defined" class="mw-redirect" title="Densely defined">Densely defined</a></li>

<li>Fredholm

<ul><li><a href="Fredholm_kernel" title="Fredholm kernel">kernel</a></li>

<li><a href="Fredholm_operator" title="Fredholm operator">operator</a></li></ul></li>

<li><a href="Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li>

<li><a href="Linear_form" title="Linear form">Functionals</a>

<ul><li><a href="Positive_linear_functional" title="Positive linear functional">positive</a></li></ul></li>

<li><a href="Pseudo-monotone_operator" title="Pseudo-monotone operator">Pseudo-monotone</a></li>

<li><a href="Normal_operator" title="Normal operator">Normal</a></li>

<li><a href="Nuclear_operator" title="Nuclear operator">Nuclear</a></li>

<li><a href="Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint</a></li>

<li><a href="Strictly_singular_operator" title="Strictly singular operator">Strictly singular</a></li>

<li><a href="Trace_class" title="Trace class">Trace class</a></li>

<li><a href="Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li>

<li><a href="Unitary_operator" title="Unitary operator">Unitary</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Operator_theory" title="Operator theory">Operator theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_algebra" title="Banach algebra">Banach algebras</a></li>

<li><a href="C*-algebra" title="C*-algebra">C*-algebras</a></li>

<li><a href="Operator_space" title="Operator space">Operator space</a></li>

<li><a href="Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">Spectrum</a>

<ul><li><a href="Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">C*-algebra</a></li>

<li><a href="Spectral_radius" title="Spectral radius">radius</a></li></ul></li>

<li><a href="Spectral_theory" title="Spectral theory">Spectral theory</a>

<li><a href="Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li></ul></li>

<li><a href="Polar_decomposition" title="Polar decomposition">Polar decomposition</a></li>

<li><a href="Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Anderson%E2%80%93Kadec_theorem" title="Anderson–Kadec theorem">Anderson–Kadec</a></li>

<li><a href="Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li>

<li><a href="Banach%E2%80%93Mazur_theorem" title="Banach–Mazur theorem">Banach–Mazur</a></li>

<li><a href="Banach%E2%80%93Saks_theorem" class="mw-redirect" title="Banach–Saks theorem">Banach–Saks</a></li>

<li><a href="Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">Banach–Schauder (open mapping)</a></li>

<li><a href="Uniform_boundedness_principle" title="Uniform boundedness principle">Banach–Steinhaus (Uniform boundedness)</a></li>

<li><a href="Bessel's_inequality" title="Bessel's inequality">Bessel's inequality</a></li>

<li><a href="Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a></li>

<li><a href="Closed_graph_theorem" title="Closed graph theorem">Closed graph</a></li>

<li><a href="Closed_range_theorem" title="Closed range theorem">Closed range</a></li>

<li><a href="Eberlein%E2%80%93%C5%A0mulian_theorem" title="Eberlein–Šmulian theorem">Eberlein–Šmulian</a></li>

<li><a href="Freudenthal_spectral_theorem" title="Freudenthal spectral theorem">Freudenthal spectral</a></li>

<li><a href="Gelfand%E2%80%93Mazur_theorem" title="Gelfand–Mazur theorem">Gelfand–Mazur</a></li>

<li><a href="Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li>

<li><a href="Goldstine_theorem" title="Goldstine theorem">Goldstine</a></li>

<li><a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a>

<ul><li><a href="Hyperplane_separation_theorem" title="Hyperplane separation theorem">hyperplane separation</a></li></ul></li>

<li><a href="Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li>

<li><a href="Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li>

<li><a href="Invariant_subspace_problem#Known_special_cases" title="Invariant subspace problem">Lomonosov's invariant subspace</a></li>

<li><a href="Mackey%E2%80%93Arens_theorem" title="Mackey–Arens theorem">Mackey–Arens</a></li>

<li><a href="Mazur's_lemma" title="Mazur's lemma">Mazur's lemma</a></li>

<li><a href="M._Riesz_extension_theorem" title="M. Riesz extension theorem">M. Riesz extension</a></li>

<li><a href="Parseval's_identity" title="Parseval's identity">Parseval's identity</a></li>

<li><a href="Riesz's_lemma" title="Riesz's lemma">Riesz's lemma</a></li>

<li><a href="Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li>

<li><a href="Ursescu_theorem#Robinson–Ursescu_theorem" title="Ursescu theorem">Robinson-Ursescu</a></li>

<li><a href="Schauder_fixed-point_theorem" title="Schauder fixed-point theorem">Schauder fixed-point</a></li>

<li><a href="Sobczyk's_theorem" title="Sobczyk's theorem">Sobczyk's theorem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Analysis</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abstract_Wiener_space" title="Abstract Wiener space">Abstract Wiener space</a></li>

<li><a href="Banach_manifold" title="Banach manifold">Banach manifold</a>

<ul><li><a href="Banach_bundle" title="Banach bundle">bundle</a></li></ul></li>

<li><a href="Bochner_space" title="Bochner space">Bochner space</a></li>

<li><a href="Convex_series" title="Convex series">Convex series</a></li>

<li><a href="Differentiation_in_Fr%C3%A9chet_spaces" title="Differentiation in Fréchet spaces">Differentiation in Fréchet spaces</a></li>

<li><a href="Derivative" title="Derivative">Derivatives</a>

<ul><li><a href="Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet</a></li>

<li><a href="Gateaux_derivative" title="Gateaux derivative">Gateaux</a></li>

<li><a href="Functional_derivative" title="Functional derivative">functional</a></li>

<li><a href="Infinite-dimensional_holomorphy" title="Infinite-dimensional holomorphy">holomorphic</a></li>

<li><a href="Quasi-derivative" title="Quasi-derivative">quasi</a></li></ul></li>

<li><a href="Integral" title="Integral">Integrals</a>

<ul><li><a href="Bochner_integral" title="Bochner integral">Bochner</a></li>

<li><a href="Dunford_integral" class="mw-redirect" title="Dunford integral">Dunford</a></li>

<li><a href="Pettis_integral" title="Pettis integral">Gelfand–Pettis</a></li>

<li><a href="Regulated_integral" title="Regulated integral">regulated</a></li>

<li><a href="Paley%E2%80%93Wiener_integral" title="Paley–Wiener integral">Paley–Wiener</a></li>

<li><a href="Functional_calculus" title="Functional calculus">Functional calculus</a>

<ul><li><a href="Borel_functional_calculus" title="Borel functional calculus">Borel</a></li>

<li><a href="Continuous_functional_calculus" title="Continuous functional calculus">continuous</a></li>

<li><a href="Holomorphic_functional_calculus" title="Holomorphic functional calculus">holomorphic</a></li></ul></li>

<li><a href="Measure_(mathematics)" title="Measure (mathematics)">Measures</a>

<ul><li><a href="Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">Lebesgue</a></li>

<li><a href="Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li>

<li><a href="Vector_measure" title="Vector measure">Vector</a></li></ul></li>

<li><a href="Weakly_measurable_function" title="Weakly measurable function">Weakly</a> / <a href="Strongly_measurable_functions" class="mw-redirect" title="Strongly measurable functions">Strongly</a> measurable function</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of sets</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Absolutely_convex_set" title="Absolutely convex set">Absolutely convex</a></li>

<li><a href="Absorbing_set" title="Absorbing set">Absorbing</a></li>

<li><a href="Affine_space" title="Affine space">Affine</a></li>

<li><a href="Balanced_set" title="Balanced set">Balanced/Circled</a></li>

<li><a href="Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Bounded</a></li>

<li><a href="Convex_set" title="Convex set">Convex</a></li>

<li><a href="Convex_cone" title="Convex cone">Convex cone (subset)</a></li>

<li><a href="Convex_series#Types_of_subsets" title="Convex series">Convex series related</a> ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx))</li>

<li><a href="Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">Linear cone (subset)</a></li>

<li><a href="Radial_set" title="Radial set">Radial</a></li>

<li><a href="Star_domain" title="Star domain">Radially convex/Star-shaped</a></li>

<li><a href="Symmetric_set" title="Symmetric set">Symmetric</a></li>

<li><a href="Zonotope" class="mw-redirect" title="Zonotope">Zonotope</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Subsets / set operations</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Affine_hull" title="Affine hull">Affine hull</a></li>

<li>(<a href="Algebraic_interior#Relative_algebraic_interior" title="Algebraic interior">Relative</a>) <a href="Algebraic_interior" title="Algebraic interior">Algebraic interior (core)</a></li>

<li><a href="Bounding_point" title="Bounding point">Bounding points</a></li>

<li><a href="Convex_hull" title="Convex hull">Convex hull</a></li>

<li><a href="Extreme_point" title="Extreme point">Extreme point</a></li>

<li><a href="Interior_(topology)" title="Interior (topology)">Interior</a></li>

<li><a href="Linear_span" title="Linear span">Linear span</a></li>

<li><a href="Minkowski_addition" title="Minkowski addition">Minkowski addition</a></li>

<li><a href="Polar_set" title="Polar set">Polar</a></li>

<li>(<a href="Algebraic_interior#Quasi_relative_interior" title="Algebraic interior">Quasi</a>) <a href="Algebraic_interior#Relative_interior" title="Algebraic interior">Relative interior</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Absolute_continuity" title="Absolute continuity">Absolute continuity AC</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ba(\Sigma )}</annotation>

</semantics>

</math><img src="./58fe61351e3531b14043fa2d09e98c2437bd1a6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.715ex; height:2.843ex;" alt="{\displaystyle ba(\Sigma )}" loading="lazy"></a></li>

<li><a href="C_space" title="C space">c space</a></li>

<li><a href="BK-space" title="BK-space">Banach coordinate BK</a></li>

<li><a href="Besov_space" title="Besov space">Besov <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p,q}^{s}(\mathbb {R} )}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{p,q}^{s}(\mathbb {R} )}</annotation>

</semantics>

</math><img src="./9919cf78ad095c237169772d2b27a37bfbef1b75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.524ex; height:3.009ex;" alt="{\displaystyle B_{p,q}^{s}(\mathbb {R} )}" loading="lazy"></a></li>

<li><a href="Birnbaum%E2%80%93Orlicz_space" class="mw-redirect" title="Birnbaum–Orlicz space">Birnbaum–Orlicz</a></li>

<li><a href="Bounded_variation" title="Bounded variation">Bounded variation BV</a></li>

<li><a href="Bs_space" title="Bs space">Bs space</a></li>

<li><a href="Continuous_functions_on_a_compact_Hausdorff_space" class="mw-redirect" title="Continuous functions on a compact Hausdorff space">Continuous C(K) with K compact Hausdorff</a></li>

<li><a href="Hardy_space" title="Hardy space">Hardy Hp</a></li>

<li><a href="Hilbert_space#Definition" title="Hilbert space">Hilbert H</a></li>

<li><a href="Morrey%E2%80%93Campanato_space" title="Morrey–Campanato space">Morrey–Campanato <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\lambda ,p}(\Omega )}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{\lambda ,p}(\Omega )}</annotation>

</semantics>

</math><img src="./8b8af58fa038369c3ec6386c6656aab82825e372.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.545ex; height:3.176ex;" alt="{\displaystyle L^{\lambda ,p}(\Omega )}" loading="lazy"></a></li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation>

</semantics>

</math><img src="./8348195cf09473662c6f59e6717722a6fc01d0f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}" loading="lazy"></a></li></ul></li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation>

</semantics>

</math><img src="./b9ab400cc4dfd865180cd84c72dc894ca457671f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}" loading="lazy"></a></li>

<li><a href="Lp_space#Weighted_Lp_spaces" title="Lp space">weighted</a></li></ul></li>

<li><a href="Schwartz_space" title="Schwartz space">Schwartz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\left(\mathbb {R} ^{n}\right)}">

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\left(\mathbb {R} ^{n}\right)}</annotation>

</semantics>

</math><img src="./0465acd58a0f31e32b095aed742d9ccc6331369c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.592ex; height:2.843ex;" alt="{\displaystyle S\left(\mathbb {R} ^{n}\right)}" loading="lazy"></a></li>

<li><a href="Segal%E2%80%93Bargmann_space" title="Segal–Bargmann space">Segal–Bargmann F</a></li>

<li><a href="Sequence_space" title="Sequence space">Sequence space</a></li>

<li><a href="Sobolev_space" title="Sobolev space">Sobolev Wk,p</a>

<ul><li><a href="Sobolev_inequality" title="Sobolev inequality">Sobolev inequality</a></li></ul></li>

<li><a href="Triebel%E2%80%93Lizorkin_space" title="Triebel–Lizorkin space">Triebel–Lizorkin</a></li>

<li><a href="Wiener_amalgam_space" title="Wiener amalgam space">Wiener amalgam <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(X,L^{p})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W(X,L^{p})}</annotation>

</semantics>

</math><img src="./b37b1dc9714960c525cb561a4828f41feb5844ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.9ex; height:2.843ex;" alt="{\displaystyle W(X,L^{p})}" loading="lazy"></a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Differential_operator" title="Differential operator">Differential operator</a></li>

<li><a href="Finite_element_method" title="Finite element method">Finite element method</a></li>

<li><a href="Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Mathematical formulation of quantum mechanics</a></li>

<li><a href="Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Ordinary Differential Equations (ODEs)</a></li>

<li><a href="Validated_numerics" title="Validated numerics">Validated numerics</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Functional_analysis_(topics_–_glossary)364" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Functional_analysis_(topics_–_glossary)364" style="font-size:114%;margin:0 4em"><a href="Functional_analysis" title="Functional analysis">Functional analysis</a> (<a href="List_of_functional_analysis_topics" title="List of functional analysis topics">topics</a> – <a href="Glossary_of_functional_analysis" title="Glossary of functional analysis">glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spaces</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_space" title="Banach space">Banach</a></li>

<li><a href="Besov_space" title="Besov space">Besov</a></li>

<li><a href="Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a></li>

<li><a href="Hilbert_space" title="Hilbert space">Hilbert</a></li>

<li><a href="H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder</a></li>

<li><a href="Nuclear_space" title="Nuclear space">Nuclear</a></li>

<li><a href="Orlicz_space" title="Orlicz space">Orlicz</a></li>

<li><a href="Schwartz_space" title="Schwartz space">Schwartz</a></li>

<li><a href="Sobolev_space" title="Sobolev space">Sobolev</a></li>

<li><a href="Topological_vector_space" title="Topological vector space">Topological vector</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Barrelled_space" title="Barrelled space">Barrelled</a></li>

<li><a href="Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li>

<li><a href="Dual_space" title="Dual space">Dual</a> (<a href="Dual_space#Algebraic_dual_space" title="Dual space">Algebraic</a> / <a href="Dual_space#Continuous_dual_space" title="Dual space">Topological</a>)</li>

<li><a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a></li>

<li><a href="Separable_space" title="Separable space">Separable</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a></li>

<li><a href="Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li>

<li><a href="Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph</a></li>

<li><a href="Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a></li>

<li><a href="Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li>

<li><a href="Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li>

<li><a href="Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li>

<li><a href="Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adjoint_operator" class="mw-redirect" title="Adjoint operator">Adjoint</a></li>

<li><a href="Bounded_operator" title="Bounded operator">Bounded</a></li>

<li><a href="Compact_operator" title="Compact operator">Compact</a></li>

<li><a href="Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li>

<li><a href="Normal_operator" title="Normal operator">Normal</a></li>

<li><a href="Nuclear_operator" title="Nuclear operator">Nuclear</a></li>

<li><a href="Trace_class" title="Trace class">Trace class</a></li>

<li><a href="Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li>

<li><a href="Unbounded_operator" title="Unbounded operator">Unbounded</a></li>

<li><a href="Unitary_operator" title="Unitary operator">Unitary</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebras</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_algebra" title="Banach algebra">Banach algebra</a></li>

<li><a href="C*-algebra" title="C*-algebra">C*-algebra</a></li>

<li><a href="Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li>

<li><a href="Operator_algebra" title="Operator algebra">Operator algebra</a></li>

<li><a href="Group_algebra_of_a_locally_compact_group" title="Group algebra of a locally compact group">Group algebra of a locally compact group</a></li>

<li><a href="Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Open problems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a></li>

<li><a href="Mahler's_conjecture" class="mw-redirect" title="Mahler's conjecture">Mahler's conjecture</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hardy_space" title="Hardy space">Hardy space</a></li>

<li><a href="Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li>

<li><a href="Heat_kernel" title="Heat kernel">Heat kernel</a></li>

<li><a href="Index_theorem" class="mw-redirect" title="Index theorem">Index theorem</a></li>

<li><a href="Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li>

<li><a href="Functional_calculus" title="Functional calculus">Functional calculus</a></li>

<li><a href="Integral_linear_operator" title="Integral linear operator">Integral linear operator</a></li>

<li><a href="Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li>

<li><a href="Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li>

<li><a href="Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li>

<li><a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li>

<li><a href="Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a> (or <a href="Generalized_function" title="Generalized function">Generalized functions</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Advanced topics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Approximation_property" title="Approximation property">Approximation property</a></li>

<li><a href="Balanced_set" title="Balanced set">Balanced set</a></li>

<li><a href="Choquet_theory" title="Choquet theory">Choquet theory</a></li>

<li><a href="Weak_topology" title="Weak topology">Weak topology</a></li>

<li><a href="Banach%E2%80%93Mazur_distance" class="mw-redirect" title="Banach–Mazur distance">Banach–Mazur distance</a></li>

<li><a href="Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li> Category</li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Topological_vector_spaces_(TVSs)267" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Topological_vector_spaces_(TVSs)267" style="font-size:114%;margin:0 4em"><a href="Topological_vector_space" title="Topological vector space">Topological vector spaces</a> (TVSs)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_space" title="Banach space">Banach space</a></li>

<li><a href="Metrizable_topological_vector_space" title="Metrizable topological vector space">Completeness</a></li>

<li><a href="Continuous_linear_operator" title="Continuous linear operator">Continuous linear operator</a></li>

<li><a href="Linear_form" title="Linear form">Linear functional</a></li>

<li><a href="Fr%C3%A9chet_space" title="Fréchet space">Fréchet space</a></li>

<li><a href="Linear_map" title="Linear map">Linear map</a></li>

<li><a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex space</a></li>

<li><a href="Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizability</a></li>

<li><a href="Operator_topologies" title="Operator topologies">Operator topologies</a></li>

<li><a href="Topological_vector_space" title="Topological vector space">Topological vector space</a></li>

<li><a href="Vector_space" title="Vector space">Vector space</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Anderson%E2%80%93Kadec_theorem" title="Anderson–Kadec theorem">Anderson–Kadec</a></li>

<li><a href="Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li>

<li><a href="Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph theorem</a></li>

<li><a href="F._Riesz's_theorem" title="F. Riesz's theorem">F. Riesz's</a></li>

<li><a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a> (<a href="Hyperplane_separation_theorem" title="Hyperplane separation theorem">hyperplane separation</a></li>

<li><a href="Vector-valued_Hahn%E2%80%93Banach_theorems" title="Vector-valued Hahn–Banach theorems">Vector-valued Hahn–Banach</a>)</li>

<li><a href="Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">Open mapping (Banach–Schauder)</a>

<ul><li><a href="Bounded_inverse_theorem" class="mw-redirect" title="Bounded inverse theorem">Bounded inverse</a></li></ul></li>

<li><a href="Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness (Banach–Steinhaus)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bilinear_operator" class="mw-redirect" title="Bilinear operator">Bilinear operator</a>

<li><a href="Linear_map" title="Linear map">Linear map</a>

<ul><li><a href="Almost_open_linear_map" class="mw-redirect" title="Almost open linear map">Almost open</a></li>

<li><a href="Bounded_operator" title="Bounded operator">Bounded</a></li>

<li><a href="Continuous_linear_operator" title="Continuous linear operator">Continuous</a></li>

<li><a href="Closed_linear_operator" title="Closed linear operator">Closed</a></li>

<li><a href="Compact_operator" title="Compact operator">Compact</a></li>

<li><a href="Densely_defined_operator" title="Densely defined operator">Densely defined</a></li>

<li><a href="Discontinuous_linear_map" title="Discontinuous linear map">Discontinuous</a></li></ul></li>

<li><a href="Topological_homomorphism" title="Topological homomorphism">Topological homomorphism</a></li>

<li><a href="Functional_(mathematics)" title="Functional (mathematics)">Functional</a>

<ul><li><a href="Linear_form" title="Linear form">Linear</a></li>

<li><a href="Bilinear_form" title="Bilinear form">Bilinear</a></li>

<li><a href="Sesquilinear_form" title="Sesquilinear form">Sesquilinear</a></li></ul></li>

<li><a href="Seminorm" title="Seminorm">Seminorm</a></li>

<li><a href="Sublinear_function" title="Sublinear function">Sublinear function</a></li>

<li><a href="Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of sets</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Absolutely_convex_set" title="Absolutely convex set">Absolutely convex/disk</a></li>

<li><a href="Absorbing_set" title="Absorbing set">Absorbing/Radial</a></li>

<li><a href="Affine_space" title="Affine space">Affine</a></li>

<li><a href="Balanced_set" title="Balanced set">Balanced/Circled</a></li>

<li><a href="Auxiliary_normed_space" title="Auxiliary normed space">Banach disks</a></li>

<li><a href="Bounding_point" title="Bounding point">Bounding points</a></li>

<li><a href="Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Bounded</a></li>

<li><a href="Complemented_subspace" title="Complemented subspace">Complemented subspace</a></li>

<li><a href="Convex_set" title="Convex set">Convex</a></li>

<li><a href="Convex_cone" title="Convex cone">Convex cone (subset)</a></li>

<li><a href="Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">Linear cone (subset)</a></li>

<li><a href="Extreme_point" title="Extreme point">Extreme point</a></li>

<li><a href="Totally_bounded_space#Topological_vector_spaces" title="Totally bounded space">Pre-compact/Totally bounded</a></li>

<li><a href="Prevalent_and_shy_sets" title="Prevalent and shy sets">Prevalent/Shy</a></li>

<li><a href="Radial_set" title="Radial set">Radial</a></li>

<li><a href="Star_domain" title="Star domain">Radially convex/Star-shaped</a></li>

<li><a href="Symmetric_set" title="Symmetric set">Symmetric</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set operations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Affine_hull" title="Affine hull">Affine hull</a></li>

<li>(<a href="Algebraic_interior#Relative_algebraic_interior" title="Algebraic interior">Relative</a>) <a href="Algebraic_interior" title="Algebraic interior">Algebraic interior (core)</a></li>

<li><a href="Convex_hull" title="Convex hull">Convex hull</a></li>

<li><a href="Linear_span" title="Linear span">Linear span</a></li>

<li><a href="Minkowski_addition" title="Minkowski addition">Minkowski addition</a></li>

<li><a href="Polar_set" title="Polar set">Polar</a></li>

<li>(<a href="Algebraic_interior#Quasi_relative_interior" title="Algebraic interior">Quasi</a>) <a href="Algebraic_interior#Relative_interior" title="Algebraic interior">Relative interior</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of TVSs</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Asplund_space" title="Asplund space">Asplund</a></li>

<li><a href="Ptak_space" title="Ptak space">B-complete/Ptak</a></li>

<li><a href="Banach_space" title="Banach space">Banach</a></li>

<li>(<a href="Countably_barrelled_space" title="Countably barrelled space">Countably</a>) <a href="Barrelled_space" title="Barrelled space">Barrelled</a></li>

<li><a href="BK-space" title="BK-space">BK-space</a></li>

<li>(<a href="Ultrabornological_space" title="Ultrabornological space">Ultra-</a>) <a href="Bornological_space" title="Bornological space">Bornological</a></li>

<li><a href="Brauner_space" title="Brauner space">Brauner</a></li>

<li><a href="Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li>

<li><a href="Convenient_vector_space" title="Convenient vector space">Convenient</a></li>

<li><a href="DF-space" title="DF-space">(DF)-space</a></li>

<li><a href="Distinguished_space" title="Distinguished space">Distinguished</a></li>

<li><a href="F-space" title="F-space">F-space</a></li>

<li><a href="FK-AK_space" title="FK-AK space">FK-AK space</a></li>

<li><a href="FK-space" title="FK-space">FK-space</a></li>

<li><a href="Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a>

<ul><li><a href="Differentiation_in_Fr%C3%A9chet_spaces#Tame_Fréchet_spaces" title="Differentiation in Fréchet spaces">tame Fréchet</a></li></ul></li>

<li><a href="Grothendieck_space" title="Grothendieck space">Grothendieck</a></li>

<li><a href="Hilbert_space" title="Hilbert space">Hilbert</a></li>

<li><a href="Infrabarreled_space" class="mw-redirect" title="Infrabarreled space">Infrabarreled</a></li>

<li><a href="Interpolation_space" title="Interpolation space">Interpolation space</a></li>

<li><a href="K-space_(functional_analysis)" title="K-space (functional analysis)">K-space</a></li>

<li><a href="LB-space" title="LB-space">LB-space</a></li>

<li><a href="LF-space" title="LF-space">LF-space</a></li>

<li><a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex space</a></li>

<li><a href="Mackey_space" title="Mackey space">Mackey</a></li>

<li><a href="Metrizable_topological_vector_space" title="Metrizable topological vector space">(Pseudo)Metrizable</a></li>

<li><a href="Montel_space" title="Montel space">Montel</a></li>

<li><a href="Quasibarrelled_space" class="mw-redirect" title="Quasibarrelled space">Quasibarrelled</a></li>

<li><a href="Quasi-complete" class="mw-redirect" title="Quasi-complete">Quasi-complete</a></li>

<li><a href="Quasinorm" title="Quasinorm">Quasinormed</a></li>

<li>(<a href="Polynomially_reflexive_space" title="Polynomially reflexive space">Polynomially</a></li>

<li><a href="Riesz_space" title="Riesz space">Riesz</a></li>

<li><a href="Schwartz_TVS" class="mw-redirect" title="Schwartz TVS">Schwartz</a></li>

<li><a href="Semi-complete" class="mw-redirect" title="Semi-complete">Semi-complete</a></li>

<li><a href="Smith_space" title="Smith space">Smith</a></li>

<li><a href="Stereotype_space" class="mw-redirect" title="Stereotype space">Stereotype</a></li>

<li><a href="Strictly_convex_space" title="Strictly convex space">Strictly</a></li>

<li><a href="Uniformly_convex_space" title="Uniformly convex space">Uniformly</a>) convex</li>

<li>(<a href="Quasi-ultrabarrelled_space" title="Quasi-ultrabarrelled space">Quasi-</a>) <a href="Ultrabarrelled_space" title="Ultrabarrelled space">Ultrabarrelled</a></li>

<li><a href="Uniformly_smooth_space" title="Uniformly smooth space">Uniformly smooth</a></li>

<li><a href="Webbed_space" title="Webbed space">Webbed</a></li>

<li><a href="Approximation_property" title="Approximation property">With the approximation property</a></li></ul>

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<ul><li> Category</li></ul>

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