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<title>Operator topologies</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Operator topologies</span></span>

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<p>In the <a href="Mathematics" title="Mathematics">mathematical</a> field of <a href="Functional_analysis" title="Functional analysis">functional analysis</a> there are several standard <a href="Topology" title="Topology">topologies</a> which are given to the algebra <span class="texhtml">B(<i>X</i>)</span> of <a href="Bounded_linear_operator" class="mw-redirect" title="Bounded linear operator">bounded linear operators</a> on a <a href="Banach_space" title="Banach space">Banach space</a> <span class="texhtml mvar" style="font-style:italic;">X</span>.

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

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<annotation encoding="application/x-tex">{\displaystyle (T_{n})_{n\in \mathbb {N} }}</annotation>

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</math></span><img src="./25fbf21de09aca7ee46e637c89b442ee86bd2fab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.887ex; height:2.843ex;" alt="{\displaystyle (T_{n})_{n\in \mathbb {N} }}" loading="lazy"></span> be a sequence of linear operators on the Banach space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span>. Consider the statement that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T_{n})_{n\in \mathbb {N} }}">

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (T_{n})_{n\in \mathbb {N} }}</annotation>

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</math></span><img src="./25fbf21de09aca7ee46e637c89b442ee86bd2fab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.887ex; height:2.843ex;" alt="{\displaystyle (T_{n})_{n\in \mathbb {N} }}" loading="lazy"></span> converges to some operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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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This could have several different meanings:

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<annotation encoding="application/x-tex">{\displaystyle \|T_{n}-T\|\to 0}</annotation>

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</math></span><img src="./f4210b587eddf0ad1a5251c564d81c9f9babad74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.154ex; height:2.843ex;" alt="{\displaystyle \|T_{n}-T\|\to 0}" loading="lazy"></span>, that is, the <a href="Operator_norm" title="Operator norm">operator norm</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}-T}">

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

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</math></span><img src="./4621ce545185442976aefc126c5c36066d080f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.053ex; height:2.509ex;" alt="{\displaystyle T_{n}-T}" loading="lazy"></span> (the supremum of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|T_{n}x-Tx\|_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle \|T_{n}x-Tx\|_{X}}</annotation>

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</math></span><img src="./e3b6baa0a78e759600869a97043db51608e9e6df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.669ex; height:2.843ex;" alt="{\displaystyle \|T_{n}x-Tx\|_{X}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span> ranges over the <a href="Unit_ball" class="mw-redirect" title="Unit ball">unit ball</a> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span>) converges to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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

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</math></span><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span>, we say that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}\to T}">

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

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</math></span><img src="./cff7bbdca89ee847341519d1e584c4d70fa01fbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.826ex; height:2.509ex;" alt="{\displaystyle T_{n}\to T}" loading="lazy"></span> in the <b><a href="Uniform_operator_topology" class="mw-redirect" title="Uniform operator topology">uniform operator topology</a></b>.</li>

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

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</math></span><img src="./b9984f231b116738e78cc3e93d3d4a36c17d292f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.486ex; height:2.509ex;" alt="{\displaystyle T_{n}x\to Tx}" loading="lazy"></span> for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span>, then we say <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}\to T}">

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</mstyle>

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

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</math></span><img src="./cff7bbdca89ee847341519d1e584c4d70fa01fbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.826ex; height:2.509ex;" alt="{\displaystyle T_{n}\to T}" loading="lazy"></span> in the <b><a href="Strong_operator_topology" title="Strong operator topology">strong operator topology</a></b>.</li>

<li>Finally, suppose that for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span> we have <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}x\to Tx}">

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

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</math></span><img src="./b9984f231b116738e78cc3e93d3d4a36c17d292f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.486ex; height:2.509ex;" alt="{\displaystyle T_{n}x\to Tx}" loading="lazy"></span> in the <a href="Weak_topology" title="Weak topology">weak topology</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span>. This means that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(T_{n}x)\to F(Tx)}">

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<annotation encoding="application/x-tex">{\displaystyle F(T_{n}x)\to F(Tx)}</annotation>

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</math></span><img src="./1c75037ca314f2f47738c80d16f4580df6b9a838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.586ex; height:2.843ex;" alt="{\displaystyle F(T_{n}x)\to F(Tx)}" loading="lazy"></span> for all continuous linear functionals <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span>. In this case we say that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}\to T}">

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</mstyle>

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

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</math></span><img src="./cff7bbdca89ee847341519d1e584c4d70fa01fbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.826ex; height:2.509ex;" alt="{\displaystyle T_{n}\to T}" loading="lazy"></span> in the <b><a href="Weak_operator_topology" title="Weak operator topology">weak operator topology</a></b>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="List_of_topologies_on_B(H)">List of topologies on B(<i>H</i>)</h2></div>

<p>There are many topologies that can be defined on <span class="texhtml">B(<i>X</i>)</span> besides the ones used above; most are at first only defined when <span class="texhtml"><i>X</i> = <i>H</i></span> is a Hilbert space, even though in many cases there are appropriate generalisations.

The topologies listed below are all locally convex, which implies that they are defined by a family of <a href="Seminorm" title="Seminorm">seminorms</a>.

</p><p>In analysis, a topology is called strong if it has many open sets and weak if it has few open sets, so that the corresponding modes of convergence are, respectively, strong and weak.

(In topology proper, these terms can suggest the opposite meaning, so strong and weak are replaced with, respectively, fine and coarse.)

The diagram on the right is a summary of the relations, with the arrows pointing from strong to weak.

</p><p>If <span class="texhtml mvar" style="font-style:italic;">H</span> is a Hilbert space, the linear space of <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> operators <span class="texhtml">B(<i>X</i>)</span> has a (unique) <a href="Predual" title="Predual">predual</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(H)_{*}}">

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<annotation encoding="application/x-tex">{\displaystyle B(H)_{*}}</annotation>

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</math></span><img src="./8caf203afca69a3e08c7d485e1ee69bdafbd267c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.691ex; height:2.843ex;" alt="{\displaystyle B(H)_{*}}" loading="lazy"></span>,

consisting of the trace class operators, whose dual is <span class="texhtml">B(<i>X</i>)</span>.

The seminorm <span class="texhtml"><i>p</i><sub><i>w</i></sub>(<i>x</i>)</span> for <i>w</i> positive in the predual is defined to be

<span class="texhtml">B(<i>w</i>, <i>x<sup>*</sup>x</i>)<sup>1/2</sup></span>.

</p><p>If <span class="texhtml mvar" style="font-style:italic;">B</span> is a vector space of linear maps on the vector space <span class="texhtml mvar" style="font-style:italic;">A</span>, then <span class="texhtml">σ(<i>A</i>, <i>B</i>)</span> is defined to be the weakest topology on <span class="texhtml mvar" style="font-style:italic;">A</span> such that all elements of <span class="texhtml mvar" style="font-style:italic;">B</span> are continuous.

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<ul><li>The <b><a href="Norm_topology" class="mw-redirect" title="Norm topology">norm topology</a></b> or <b>uniform topology</b> or <b>uniform operator topology</b> is defined by the usual norm ||<i>x</i>|| on <span class="texhtml">B(<i>H</i>)</span>. It is stronger than all the other topologies below.</li>

<li>The <b><a href="Weak_topology" title="Weak topology">weak (Banach space) topology</a></b> is <span class="texhtml">σ(B(<i>H</i>), B(<i>H</i>)<sup>*</sup>)</span>, in other words the weakest topology such that all elements of the dual <span class="texhtml">B(<i>H</i>)<sup>*</sup></span> are continuous. It is the weak topology on the Banach space <span class="texhtml">B(<i>H</i>)</span>. It is stronger than the ultraweak and weak operator topologies. (Warning: the weak Banach space topology and the weak operator topology and the ultraweak topology are all sometimes called the weak topology, but they are different.)</li>

<li>The <b><a href="Mackey_topology" title="Mackey topology">Mackey topology</a></b> or <b>Arens-Mackey topology</b> is the strongest locally convex topology on <span class="texhtml">B(<i>H</i>)</span> such that the dual is <span class="texhtml">B(<i>H</i>)<sub>*</sub></span>, and is also the uniform convergence topology on <span class="texhtml">Bσ(B(<i>H</i>)<sub>*</sub></span>, <span class="texhtml">B(<i>H</i>)</span>-compact convex subsets of <span class="texhtml">B(<i>H</i>)<sub>*</sub></span>. It is stronger than all topologies below.</li>

<li>The <b>σ-strong-<sup>*</sup> topology</b> or <b>ultrastrong-<sup>*</sup> topology</b> is the weakest topology stronger than the ultrastrong topology such that the adjoint map is continuous. It is defined by the family of seminorms <span class="texhtml"><i>p</i><sub><i>w</i></sub>(<i>x</i>)</span> and <span class="texhtml"><i>p</i><sub><i>w</i></sub>(<i>x</i><sup>*</sup>)</span> for positive elements <span class="texhtml mvar" style="font-style:italic;">w</span> of <span class="texhtml">B(<i>H</i>)<sub>*</sub></span>. It is stronger than all topologies below.</li>

<li>The <b>σ-strong topology</b> or <b><a href="Ultrastrong_topology" title="Ultrastrong topology">ultrastrong topology</a></b> or <b>strongest topology</b> or <b>strongest operator topology</b> is defined by the family of seminorms <span class="texhtml"><i>p</i><sub><i>w</i></sub>(<i>x</i>)</span> for positive elements <span class="texhtml mvar" style="font-style:italic;">w</span> of <span class="texhtml">B(<i>H</i>)<sub>*</sub></span>. It is stronger than all the topologies below other than the strong<sup>*</sup> topology. Warning: in spite of the name "strongest topology", it is weaker than the norm topology.)</li>

<li>The <b>σ-weak topology</b> or <b>ultraweak topology</b> or <b><a href="Weak-star_operator_topology" class="mw-redirect" title="Weak-star operator topology">weak-<sup>*</sup> operator topology</a></b> or <b>weak-* topology</b> or <b>weak topology</b> or <b><span class="texhtml">σ(B(<i>H</i>), B(<i>H</i>)<sub>*</sub></span>) topology</b> is defined by the family of seminorms |(<i>w</i>, <i>x</i>)| for elements <i>w</i> of <span class="texhtml">B(<i>H</i>)<sub>*</sub></span>. It is stronger than the weak operator topology. (Warning: the weak Banach space topology and the weak operator topology and the ultraweak topology are all sometimes called the weak topology, but they are different.)</li>

<li>The <b>strong-<sup>*</sup> operator topology</b> or <b>strong-<sup>*</sup> topology</b> is defined by the seminorms ||<i>x</i>(<i>h</i>)|| and ||<i>x</i><sup>*</sup>(<i>h</i>)|| for <span class="texhtml"><i>h</i> ∈ <i>H</i></span>. It is stronger than the strong and weak operator topologies.</li>

<li>The <b><a href="Strong_operator_topology" title="Strong operator topology">strong operator topology</a></b> (SOT) or <b>strong topology</b> is defined by the seminorms ||<i>x</i>(<i>h</i>)|| for <span class="texhtml"><i>h</i> ∈ <i>H</i></span>. It is stronger than the weak operator topology.</li>

<li>The <b><a href="Weak_operator_topology" title="Weak operator topology">weak operator topology</a></b> (WOT) or <b>weak topology</b> is defined by the seminorms |(<i>x</i>(<i>h</i><sub>1</sub>), <i>h</i><sub>2</sub>)| for <span class="texhtml"><i>h</i><sub>1</sub>, <i>h</i><sub>2</sub> ∈ <i>H</i></span>. (Warning: the weak Banach space topology, the weak operator topology, and the ultraweak topology are all sometimes called the weak topology, but they are different.)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Relations_between_the_topologies">Relations between the topologies</h2></div>

<p>The continuous linear functionals on <span class="texhtml">B(<i>H</i>)</span> for the weak, strong, and strong<sup>*</sup> (operator) topologies are the same, and are the finite linear combinations of the linear functionals

(x<i>h</i><sub>1</sub>, <i>h</i><sub>2</sub>) for <span class="texhtml"><i>h</i><sub>1</sub>, <i>h</i><sub>2</sub> ∈ <i>H</i></span>.

The continuous linear functionals on <span class="texhtml">B(<i>H</i>)</span> for the ultraweak, ultrastrong, ultrastrong<sup>*</sup> and Arens-Mackey topologies are the same, and are the elements of the predual <span class="texhtml">B(<i>H</i>)<sub>*</sub></span>.

</p><p>By definition, the continuous linear functionals in the norm topology are the same as those in the weak Banach space topology.

This dual is a rather large space with many pathological elements.

</p><p>On norm bounded sets of <span class="texhtml">B(<i>H</i>)</span>, the weak (operator) and ultraweak topologies coincide. This can be seen via, for instance, the <a href="Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu theorem</a>.

For essentially the same reason, the ultrastrong

topology is the same as the strong topology on any (norm) bounded subset of <span class="texhtml">B(<i>H</i>)</span>.

Same is true for the Arens-Mackey topology, the ultrastrong<sup>*</sup>, and the strong<sup>*</sup> topology.

</p><p>In locally convex spaces, closure of convex sets can be characterized by the continuous linear functionals. Therefore, for a <a href="Convex_set" title="Convex set">convex</a> subset <span class="texhtml mvar" style="font-style:italic;">K</span> of <span class="texhtml">B(<i>H</i>)</span>, the conditions that <span class="texhtml mvar" style="font-style:italic;">K</span> be closed in the ultrastrong<sup>*</sup>, ultrastrong, and ultraweak topologies are all equivalent and are also equivalent to the conditions that

for all <span class="texhtml"><i>r</i> > 0</span>, <span class="texhtml mvar" style="font-style:italic;">K</span> has closed intersection with the closed ball of radius <span class="texhtml mvar" style="font-style:italic;">r</span> in the strong<sup>*</sup>, strong, or weak (operator) topologies.

</p><p>The norm topology is metrizable and the others are not; in fact they fail to be <a href="First-countable" class="mw-redirect" title="First-countable">first-countable</a>.

However, when <span class="texhtml mvar" style="font-style:italic;">H</span> is separable, all the topologies above are metrizable when restricted to the unit ball (or to any norm-bounded subset).

</p>

<div class="mw-heading mw-heading2"><h2 id="Topology_to_use">Topology to use</h2></div>

<p>The most commonly used topologies are the norm, strong, and weak operator topologies.

The weak operator topology is useful for compactness arguments, because the unit ball is compact by the <a href="Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu theorem</a>.

The norm topology is fundamental because it makes <span class="texhtml">B(<i>H</i>)</span> into a Banach space, but it is too strong for many purposes; for example, <span class="texhtml">B(<i>H</i>)</span> is not separable in this topology.

The strong operator topology could be the most commonly used.

</p><p>The ultraweak and ultrastrong topologies are better-behaved than the weak and strong operator topologies, but their definitions are more complicated, so they are usually not used unless their better properties are really needed.

For example, the dual space of <span class="texhtml">B(<i>H</i>)</span> in the weak or strong operator topology is too small to have much analytic content.

</p><p>The adjoint map is not continuous in the strong operator and ultrastrong topologies, while the strong* and ultrastrong* topologies are modifications so that the adjoint becomes continuous. They are not used very often.

</p><p>The Arens–Mackey topology and the weak Banach space topology are relatively rarely used.

</p><p>To summarize, the three essential topologies on <span class="texhtml">B(<i>H</i>)</span> are the norm, ultrastrong, and ultraweak topologies.

The weak and strong operator topologies are widely used as convenient approximations to the ultraweak and ultrastrong topologies. The other topologies are relatively obscure.

</p>

<ul><li><a href="Bounded_operator" title="Bounded operator">Bounded operator</a></li>

<li><a href="Continuous_linear_operator" title="Continuous linear operator">Continuous linear operator</a> – Function between topological vector spaces</li>

<li><a href="Hilbert_space" title="Hilbert space">Hilbert space</a> – Type of vector space in math</li>

<li><a href="List_of_topologies" title="List of topologies">List of topologies</a> – List of concrete topologies and topological spaces</li>

<li><a href="Modes_of_convergence" title="Modes of convergence">Modes of convergence</a> – Property of a sequence or series</li>

<li><a href="Norm_(mathematics)" title="Norm (mathematics)">Norm (mathematics)</a> – Length in a vector space</li>

<li><a href="Topologies_on_spaces_of_linear_maps" title="Topologies on spaces of linear maps">Topologies on spaces of linear maps</a></li>

<li><a href="Vague_topology" title="Vague topology">Vague topology</a></li>

<li><a href="Weak_convergence_(Hilbert_space)" title="Weak convergence (Hilbert space)">Weak convergence (Hilbert space)</a> – Type of convergence in Hilbert spaces</li></ul>

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

<ul><li><i>Functional analysis</i>, by Reed and Simon, <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-585050-6</bdi></li>

<li><i>Theory of Operator Algebras I</i>, by M. Takesaki (especially chapter II.2) <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-42248-X</bdi></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>) <a href="Reflexive_space" title="Reflexive space">Reflexive</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="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 <span style="font-size: 85%;">(subset)</span></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, (H<i>x</i>), and (Hw<i>x</i>))</li>

<li><a href="Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">Linear cone <span style="font-size: 85%;">(subset)</span></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 <i>AC</i></a></li>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span></a></li>

<li><a href="C_space" title="C space">c space</a></li>

<li><a href="BK-space" title="BK-space">Banach coordinate <i>BK</i></a></li>

<li><a href="Besov_space" title="Besov space">Besov <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span></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 <i>BV</i></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 <i>C(K)</i> with <i>K</i> compact Hausdorff</a></li>

<li><a href="Hardy_space" title="Hardy space">Hardy H<sup><i>p</i></sup></a></li>

<li><a href="Hilbert_space#Definition" title="Hilbert space">Hilbert <i>H</i></a></li>

<li><a href="Morrey%E2%80%93Campanato_space" title="Morrey–Campanato space">Morrey–Campanato <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span></a></li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span></a></li></ul></li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span></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 <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span></a></li>

<li><a href="Segal%E2%80%93Bargmann_space" title="Segal–Bargmann space">Segal–Bargmann <i>F</i></a></li>

<li><a href="Sequence_space" title="Sequence space">Sequence space</a></li>

<li><a href="Sobolev_space" title="Sobolev space">Sobolev W<sup><i>k,p</i></sup></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 <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span></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="Hilbert_spaces69" 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="Hilbert_spaces69" style="font-size:114%;margin:0 4em"><a href="Hilbert_space" title="Hilbert space">Hilbert spaces</a></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="Hermitian_adjoint" title="Hermitian adjoint">Adjoint</a></li>

<li><a href="Inner_product_space" title="Inner product space">Inner product</a> and <a href="L-semi-inner_product" title="L-semi-inner product">L-semi-inner product</a></li>

<li><a href="Hilbert_space" title="Hilbert space">Hilbert space</a> and <a href="Prehilbert_space" class="mw-redirect" title="Prehilbert space">Prehilbert space</a></li>

<li><a href="Orthogonal_complement" title="Orthogonal complement">Orthogonal complement</a></li>

<li><a href="Orthonormal_basis" title="Orthonormal basis">Orthonormal basis</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="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="Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other results</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="Hilbert_projection_theorem" title="Hilbert projection theorem">Hilbert projection theorem</a></li>

<li><a href="Parseval's_identity" title="Parseval's identity">Parseval's identity</a></li>

<li><a href="Polarization_identity" title="Polarization identity">Polarization identity</a> (<a href="Parallelogram_law#The_parallelogram_law_in_inner_product_spaces" title="Parallelogram law">Parallelogram law</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-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">Compact operator on Hilbert space</a></li>

<li><a href="Densely_defined_operator" title="Densely defined operator">Densely defined</a></li>

<li><a href="Sesquilinear_form#Hermitian_form" title="Sesquilinear form">Hermitian form</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="Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint</a></li>

<li><a href="Sesquilinear_form" title="Sesquilinear form">Sesquilinear form</a></li>

<li><a href="Trace_class" title="Trace class">Trace class</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%">Examples</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="Distribution_(mathematics)" title="Distribution (mathematics)"><i>C</i><sup><i>n</i></sup>(<i>K</i>) with <i>K</i> compact & <i>n</i><∞</a></li>

<li><a href="Segal%E2%80%93Bargmann_space" title="Segal–Bargmann space">Segal–Bargmann <i>F</i></a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Duality_and_spaces_of_linear_maps129" 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="Duality_and_spaces_of_linear_maps129" style="font-size:114%;margin:0 4em"><a href="Dual_system" title="Dual system">Duality</a> and spaces of <a href="Linear_operator" class="mw-redirect" title="Linear operator">linear</a> maps</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="Dual_space" title="Dual space">Dual space</a></li>

<li><a href="Dual_system" title="Dual system">Dual system</a></li>

<li><a href="Dual_topology" title="Dual topology">Dual topology</a></li>

<li><a href="Duality_(mathematics)" title="Duality (mathematics)">Duality</a></li>

<li><a href="Polar_set" title="Polar set">Polar set</a></li>

<li><a href="Polar_topology" title="Polar topology">Polar topology</a></li>

<li><a href="Topologies_on_spaces_of_linear_maps" title="Topologies on spaces of linear maps">Topologies on spaces of linear maps</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"></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="Norm_topology" class="mw-redirect" title="Norm topology">Norm topology</a>

<li><a href="Ultraweak_topology" title="Ultraweak topology">Ultraweak/Weak-*</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>

<li><a href="Weak_convergence_(Hilbert_space)" title="Weak convergence (Hilbert space)">in Hilbert spaces</a></li></ul></li>

<li><a href="Mackey_topology" title="Mackey topology">Mackey </a></li>

<li><a href="Strong_dual_space" title="Strong dual space">Strong dual</a>

<ul><li><a href="Strong_topology_(polar_topology)" class="mw-redirect" title="Strong topology (polar topology)">polar topology</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></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-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li>

<li><a href="Mackey%E2%80%93Arens_theorem" title="Mackey–Arens theorem">Mackey–Arens</a></li></ul>

<ul><li><a href="Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose of a linear map</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Subsets</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="Saturated_family" title="Saturated family">Saturated family</a></li>

<li><a href="Total_set" title="Total set">Total set</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other concepts</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="Biorthogonal_system" title="Biorthogonal system">Biorthogonal system</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="Reflexive_space" title="Reflexive space">Reflexive</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><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> 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="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>

<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 <span style="font-size: 85%;">(subset)</span></a></li>

<li><a href="Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">Linear cone <span style="font-size: 85%;">(subset)</span></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>

</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="Semi-reflexive_space" title="Semi-reflexive space">Semi-</a>) <a href="Reflexive_space" title="Reflexive space">Reflexive</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>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-03-03" href="https://en.wikipedia.org/wiki/?title=Operator_topologies&oldid=1278600002">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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