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<title>Discontinuous linear map</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Discontinuous linear map</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, <a href="Linear_map" title="Linear map">linear maps</a> form an important class of "simple" <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> which preserve the algebraic structure of <a href="Linear_space" class="mw-redirect" title="Linear space">linear spaces</a> and are often used as approximations to more general functions (see <a href="Linear_approximation" title="Linear approximation">linear approximation</a>). If the spaces involved are also <a href="Topological_space" title="Topological space">topological spaces</a> (that is, <a href="Topological_vector_space" title="Topological vector space">topological vector spaces</a>), then it makes sense to ask whether all linear maps are <a href="Continuous_map" class="mw-redirect" title="Continuous map">continuous</a>. It turns out that for maps defined on infinite-<a href="Dimension_(linear_algebra)" class="mw-redirect" title="Dimension (linear algebra)">dimensional</a> topological vector spaces (e.g., infinite-dimensional <a href="Normed_space" class="mw-redirect" title="Normed space">normed spaces</a>), the answer is generally no: there exist <b>discontinuous linear maps</b>. If the domain of definition is <a href="Complete_space" class="mw-redirect" title="Complete space">complete</a>, it is trickier; such maps can be proven to exist, but the proof relies on the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a> and does not provide an explicit example.

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<div class="mw-heading mw-heading2"><h2 id="A_linear_map_from_a_finite-dimensional_space_is_always_continuous">A linear map from a finite-dimensional space is always continuous</h2></div>

<p>Let <i>X</i> and <i>Y</i> be two normed spaces and <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:X\to Y}">

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

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</math></span><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"></span> a linear map from <i>X</i> to <i>Y</i>. If <i>X</i> is <a href="Finite-dimensional" class="mw-redirect" title="Finite-dimensional">finite-dimensional</a>, choose a basis <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 \left(e_{1},e_{2},\ldots ,e_{n}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \left(e_{1},e_{2},\ldots ,e_{n}\right)}</annotation>

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</math></span><img src="./a2f629f4cfaa29ec0f79e9c6182b8adb3ec931e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.599ex; height:2.843ex;" alt="{\displaystyle \left(e_{1},e_{2},\ldots ,e_{n}\right)}" loading="lazy"></span> in <i>X</i> which may be taken to be unit vectors. Then,

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<annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{i=1}^{n}x_{i}f(e_{i}),}</annotation>

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and so by the <a href="Triangle_inequality" title="Triangle inequality">triangle inequality</a>,

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<annotation encoding="application/x-tex">{\displaystyle \|f(x)\|=\left\|\sum _{i=1}^{n}x_{i}f(e_{i})\right\|\leq \sum _{i=1}^{n}|x_{i}|\|f(e_{i})\|.}</annotation>

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Letting

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<annotation encoding="application/x-tex">{\displaystyle M=\sup _{i}\{\|f(e_{i})\|\},}</annotation>

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and using the fact that

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}|x_{i}|\leq C\|x\|}</annotation>

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for some <i>C</i>>0 which follows from the fact that <a href="Norm_(mathematics)#Properties" title="Norm (mathematics)">any two norms on a finite-dimensional space are equivalent</a>, one finds

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<annotation encoding="application/x-tex">{\displaystyle \|f(x)\|\leq \left(\sum _{i=1}^{n}|x_{i}|\right)M\leq CM\|x\|.}</annotation>

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Thus, <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="./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"></span> is a <a href="Bounded_linear_operator" class="mw-redirect" title="Bounded linear operator">bounded linear operator</a> and so is continuous. In fact, to see this, simply note that <i>f</i> is linear,

and therefore <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(x)-f(x')\|=\|f(x-x')\|\leq K\|x-x'\|}">

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<annotation encoding="application/x-tex">{\displaystyle \|f(x)-f(x')\|=\|f(x-x')\|\leq K\|x-x'\|}</annotation>

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</math></span><img src="./dd2fd95a080259b1be6ecf772d9a35c93744fad3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.055ex; height:3.009ex;" alt="{\displaystyle \|f(x)-f(x')\|=\|f(x-x')\|\leq K\|x-x'\|}" loading="lazy"></span> for some universal constant <i>K</i>. Thus for any <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 \epsilon >0,}">

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

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</math></span><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"></span>

we can choose <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 \delta \leq \epsilon /K}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \delta \leq \epsilon /K}</annotation>

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</math></span><img src="./a9e6f350cb0f0194b986d268e606b80156db35ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.32ex; height:2.843ex;" alt="{\displaystyle \delta \leq \epsilon /K}" loading="lazy"></span> so 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(B(x,\delta ))\subseteq B(f(x),\epsilon )}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(B(x,\delta ))\subseteq B(f(x),\epsilon )}</annotation>

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</math></span><img src="./bcb9d71ce89cb085a02a896124cf3ba5991554a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.141ex; height:2.843ex;" alt="{\displaystyle f(B(x,\delta ))\subseteq B(f(x),\epsilon )}" loading="lazy"></span> (<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(x,\delta )}">

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

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</math></span><img src="./59e252e430b1c641ae14989fbcc4065f4848977b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.986ex; height:2.843ex;" alt="{\displaystyle B(x,\delta )}" loading="lazy"></span> and

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<annotation encoding="application/x-tex">{\displaystyle B(f(x),\epsilon )}</annotation>

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</math></span><img src="./3dc39048f6a7cec3041acb667a27f7905eb97135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.969ex; height:2.843ex;" alt="{\displaystyle B(f(x),\epsilon )}" loading="lazy"></span> are the normed balls around <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> and <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(x)}">

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

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</math></span><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span>), which gives continuity.

</p><p>If <i>X</i> is infinite-dimensional, this proof will fail as there is no guarantee that the <a href="Supremum" class="mw-redirect" title="Supremum">supremum</a> <i>M</i> exists. If <i>Y</i> is the zero space {0}, the only map between <i>X</i> and <i>Y</i> is the zero map which is trivially continuous. In all other cases, when <i>X</i> is infinite-dimensional and <i>Y</i> is not the zero space, one can find a discontinuous map from <i>X</i> to <i>Y</i>.

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<div class="mw-heading mw-heading2"><h2 id="A_concrete_example">A concrete example</h2></div>

<p>Examples of discontinuous linear maps are easy to construct in spaces that are not complete; on any Cauchy sequence <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 e_{i}}">

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

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</math></span><img src="./ebdc3a9cb1583d3204eff8918b558c293e0d2cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}" loading="lazy"></span> of linearly independent vectors which does not have a limit, there is a linear 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> such that the quantities <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(e_{i})\|/\|e_{i}\|}">

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<annotation encoding="application/x-tex">{\displaystyle \|T(e_{i})\|/\|e_{i}\|}</annotation>

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</math></span><img src="./7472b47154d40ac3ffbed182caf0d45fbc3c248f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.024ex; height:2.843ex;" alt="{\displaystyle \|T(e_{i})\|/\|e_{i}\|}" loading="lazy"></span> grow without bound. In a sense, the linear operators are not continuous because the space has "holes".

</p><p>For example, consider the 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> of real-valued <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth functions</a> on the interval [0, 1] with the <a href="Uniform_norm" title="Uniform norm">uniform norm</a>, that is,

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<annotation encoding="application/x-tex">{\displaystyle \|f\|=\sup _{x\in [0,1]}|f(x)|.}</annotation>

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The <i><a href="Derivative" title="Derivative">derivative</a>-at-a-point</i> map, given by

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<annotation encoding="application/x-tex">{\displaystyle T(f)=f'(0)\,}</annotation>

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defined 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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</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> and with real values, is linear, but not continuous. Indeed, consider the sequence

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<annotation encoding="application/x-tex">{\displaystyle f_{n}(x)={\frac {\sin(n^{2}x)}{n}}}</annotation>

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for <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 n\geq 1}">

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

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</math></span><img src="./d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}" loading="lazy"></span>. This sequence converges uniformly to the constantly zero function, but

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<annotation encoding="application/x-tex">{\displaystyle T(f_{n})={\frac {n^{2}\cos(n^{2}\cdot 0)}{n}}=n\to \infty }</annotation>

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</p><p>as <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 n\to \infty }">

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

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</math></span><img src="./a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }" loading="lazy"></span> instead 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(f_{n})\to T(0)=0}">

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<annotation encoding="application/x-tex">{\displaystyle T(f_{n})\to T(0)=0}</annotation>

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</math></span><img src="./9048c35a2a281437d63c762be6c927a953a28cad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.286ex; height:2.843ex;" alt="{\displaystyle T(f_{n})\to T(0)=0}" loading="lazy"></span>, as would hold for a continuous map. Note 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}">

</mstyle>

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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> is real-valued, and so is actually a <a href="Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a> 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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</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> (an element of the algebraic <a href="Dual_space" title="Dual space">dual space</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 X^{*}}">

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

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</math></span><img src="./01924e6e5570e2631081fea6c6981b4872d3e04b.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^{*}}" loading="lazy"></span>). The linear map <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\to X}">

</mstyle>

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

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</math></span><img src="./83d8a6029587ee9b365bdeab1e2f4b7c469b0219.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.574ex; height:2.176ex;" alt="{\displaystyle X\to X}" loading="lazy"></span> which assigns to each function its derivative is similarly discontinuous. Note that although the derivative operator is not continuous, it is <a href="Closed_operator" class="mw-redirect" title="Closed operator">closed</a>.

</p><p>The fact that the domain is not complete here is important: discontinuous operators on complete spaces require a little more work.

</p>

<div class="mw-heading mw-heading2"><h2 id="A_nonconstructive_example">A nonconstructive example</h2></div>

<p>An algebraic basis for the <a href="Real_number" title="Real number">real numbers</a> as a vector space over the <a href="Rationals" class="mw-redirect" title="Rationals">rationals</a> is known as a <a href="Hamel_basis" class="mw-redirect" title="Hamel basis">Hamel basis</a> (note that some authors use this term in a broader sense to mean an algebraic basis of <i>any</i> vector space). Note that any two <a href="Commensurability_(mathematics)" title="Commensurability (mathematics)">noncommensurable</a> numbers, say 1 and <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 \pi }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"></span>, are linearly independent. One may find a Hamel basis containing them, and define a map <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:\mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1e3a10a3ad05781f5cf9c2d875a02227e21a8448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }" loading="lazy"></span> so 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(\pi )=0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\pi )=0,}</annotation>

</semantics>

</math></span><img src="./d0c8beb8efc21b353350163d6bcab7717fd44851.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.328ex; height:2.843ex;" alt="{\displaystyle f(\pi )=0,}" loading="lazy"></span> <i>f</i> acts as the identity on the rest of the Hamel basis, and extend to all 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 \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> by linearity. Let {<i>r</i><sub><i>n</i></sub>}<sub><i>n</i></sub> be any sequence of rationals which 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 \pi }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"></span>. Then lim<sub><i>n</i></sub> <i>f</i>(<i>r</i><sub><i>n</i></sub>) = π, but <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(\pi )=0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\pi )=0.}</annotation>

</semantics>

</math></span><img src="./badc9636370e77730dc111a13c4304dc3bb64fa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.328ex; height:2.843ex;" alt="{\displaystyle f(\pi )=0.}" loading="lazy"></span> By construction, <i>f</i> is linear over <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 \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"></span> (not over <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 \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>), but not continuous. Note that <i>f</i> is also not <a href="Measurable_function" title="Measurable function">measurable</a>; an <a href="Additive_map" title="Additive map">additive</a> real function is linear if and only if it is measurable, so for every such function there is a <a href="Vitali_set" title="Vitali set">Vitali set</a>. The construction of <i>f</i> relies on the axiom of choice.

</p><p>This example can be extended into a general theorem about the existence of discontinuous linear maps on any infinite-dimensional normed space (as long as the codomain is not trivial).

</p>

<div class="mw-heading mw-heading2"><h2 id="General_existence_theorem">General existence theorem</h2></div>

<p>Discontinuous linear maps can be proven to exist more generally, even if the space is complete. Let <i>X</i> and <i>Y</i> be <a href="Normed_space" class="mw-redirect" title="Normed space">normed spaces</a> over the field <i>K</i> 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 K=\mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a6419d3aa99701ca996737b17a5e1174d53e6c9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.842ex; height:2.176ex;" alt="{\displaystyle K=\mathbb {R} }" loading="lazy"></span> or <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 K=\mathbb {C} .}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./039c37276bbbaf7accc9a2395aeaf1da9ea5db25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.489ex; height:2.176ex;" alt="{\displaystyle K=\mathbb {C} .}" loading="lazy"></span> Assume that <i>X</i> is infinite-dimensional and <i>Y</i> is not the zero space. We will find a discontinuous linear map <i>f</i> from <i>X</i> to <i>K</i>, which will imply the existence of a discontinuous linear map <i>g</i> from <i>X</i> to <i>Y</i> given by the formula <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 g(x)=f(x)y_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)=f(x)y_{0}}</annotation>

</semantics>

</math></span><img src="./2f8429e3a2ebedb45ac8e90e88716dc7f38ea2e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.964ex; height:2.843ex;" alt="{\displaystyle g(x)=f(x)y_{0}}" 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 y_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./6d943dbbb0b56ca750c4d62c5b54b4ae29a773da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{0}}" loading="lazy"></span> is an arbitrary nonzero vector in <i>Y</i>.

</p><p>If <i>X</i> is infinite-dimensional, to show the existence of a linear functional which is not continuous then amounts to constructing <i>f</i> which is not bounded. For that, consider a <a href="Sequence" title="Sequence">sequence</a> (<i>e</i><sub><i>n</i></sub>)<sub><i>n</i></sub> (<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 n\geq 1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}" loading="lazy"></span>) of <a href="Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> vectors in <i>X</i>, which we normalize. Then, we define

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span></span>

for each <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 n=1,2,\ldots }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./3e6fe6c942f7596c0d3ade1a83faab2e290d0927.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.609ex; height:2.509ex;" alt="{\displaystyle n=1,2,\ldots }" loading="lazy"></span> Complete this sequence of linearly independent vectors to a <a href="Basis_(vector_space)" class="mw-redirect" title="Basis (vector space)">vector space basis</a> of <i>X</i> by defining <i>T</i> at the other vectors in the basis to be zero. <i>T</i> so defined will extend uniquely to a linear map on <i>X</i>, and since it is clearly not bounded, it is not continuous.

</p><p>Notice that by using the fact that any set of linearly independent vectors can be completed to a basis, we implicitly used the axiom of choice, which was not needed for the concrete example in the previous section.

</p>

<div class="mw-heading mw-heading2"><h2 id="Role_of_the_axiom_of_choice">Role of the axiom of choice</h2></div>

<p>As noted above, the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a> (AC) is used in the general existence theorem of discontinuous linear maps. In fact, there are no constructive examples of discontinuous linear maps with complete domain (for example, <a href="Banach_space" title="Banach space">Banach spaces</a>). In analysis as it is usually practiced by working mathematicians, the axiom of choice is always employed (it is an axiom of <a href="ZFC" class="mw-redirect" title="ZFC">ZFC</a> <a href="Set_theory" title="Set theory">set theory</a>); thus, to the analyst, all infinite-dimensional topological vector spaces admit discontinuous linear maps.

</p><p>On the other hand, in 1970 <a href="Robert_M._Solovay" title="Robert M. Solovay">Robert M. Solovay</a> exhibited a <a href="Model_(model_theory)" class="mw-redirect" title="Model (model theory)">model</a> of <a href="Set_theory" title="Set theory">set theory</a> in which every set of reals is measurable.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This implies that there are no discontinuous linear real functions. Clearly AC does not hold in the model.

</p><p>Solovay's result shows that it is not necessary to assume that all infinite-dimensional vector spaces admit discontinuous linear maps, and there are schools of analysis which adopt a more <a href="Constructivism_(mathematics)" class="mw-redirect" title="Constructivism (mathematics)">constructivist</a> viewpoint. For example, H. G. Garnir, in searching for so-called "dream spaces" (topological vector spaces on which every linear map into a normed space is continuous), was led to adopt ZF + <a href="Dependent_choice" class="mw-redirect" title="Dependent choice">DC</a> + <a href="Baire_property" class="mw-redirect" title="Baire property">BP</a> (dependent choice is a weakened form and the <a href="Baire_property" class="mw-redirect" title="Baire property">Baire property</a> is a negation of strong AC) as his axioms to prove the Garnir–Wright closed graph theorem which states, among other things, that any linear map from an <a href="F-space" title="F-space">F-space</a> to a TVS is continuous. Going to the extreme of <a href="Constructivism_(mathematics)" class="mw-redirect" title="Constructivism (mathematics)">constructivism</a>, there is Ceitin's theorem, which states that <i>every</i> function is continuous (this is to be understood in the terminology of constructivism, according to which only representable functions are considered to be functions).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Such stances are held by only a small minority of working mathematicians.

</p><p>The upshot is that the existence of discontinuous linear maps depends on AC; it is consistent with set theory without AC that there are no discontinuous linear maps on complete spaces. In particular, no concrete construction such as the derivative can succeed in defining a discontinuous linear map everywhere on a complete space.

</p>

<div class="mw-heading mw-heading2"><h2 id="Closed_operators">Closed operators</h2></div>

<p>Many naturally occurring linear discontinuous operators are <a href="Closed_operator" class="mw-redirect" title="Closed operator">closed</a>, a class of operators which share some of the features of continuous operators. It makes sense to ask which linear operators on a given space are closed. The <a href="Closed_graph_theorem" title="Closed graph theorem">closed graph theorem</a> asserts that an <i>everywhere-defined</i> closed operator on a complete domain is continuous, so to obtain a discontinuous closed operator, one must permit operators which are not defined everywhere.

</p><p>To be more concrete, let <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}">

</mstyle>

</mrow>

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

</semantics>

</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> be a map from <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>

</semantics>

</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> 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 Y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> with domain <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 \operatorname {Dom} (T),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Dom} (T),}</annotation>

</semantics>

</math></span><img src="./dde0fe7b4afd29f5ed467a755603dab8aed8ff66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.966ex; height:2.843ex;" alt="{\displaystyle \operatorname {Dom} (T),}" loading="lazy"></span> written <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:\operatorname {Dom} (T)\subseteq X\to Y.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T:\operatorname {Dom} (T)\subseteq X\to Y.}</annotation>

</semantics>

</math></span><img src="./2e7da62c0c00f4fa157bc5e9fc3fc114435e476d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.006ex; height:2.843ex;" alt="{\displaystyle T:\operatorname {Dom} (T)\subseteq X\to Y.}" loading="lazy"></span> We don't lose much if we replace <i>X</i> by the closure 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 \operatorname {Dom} (T).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Dom} (T).}</annotation>

</semantics>

</math></span><img src="./248b242bfa20b52727465a9aec6d07127e508962.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.966ex; height:2.843ex;" alt="{\displaystyle \operatorname {Dom} (T).}" loading="lazy"></span> That is, in studying operators that are not everywhere-defined, one may restrict one's attention to <a href="Densely_defined_operator" title="Densely defined operator">densely defined operators</a> without loss of generality.

</p><p>If the graph <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 \Gamma (T)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./5ba264df59493ce784931ebb1b42f14110174b78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.898ex; height:2.843ex;" alt="{\displaystyle \Gamma (T)}" loading="lazy"></span> 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}">

</mstyle>

</mrow>

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

</semantics>

</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> is closed 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\times Y,}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c012108ac139316994be07f2ecf2897382db40f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.241ex; height:2.509ex;" alt="{\displaystyle X\times Y,}" loading="lazy"></span> we call <i>T</i> <i>closed</i>. Otherwise, consider its closure <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 {\overline {\Gamma (T)}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d8634290b858ef49e47e1b3114bf77ca418952dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.013ex; height:3.676ex;" alt="{\displaystyle {\overline {\Gamma (T)}}}" loading="lazy"></span> 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\times Y.}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./41f39ef78be28dc2b9015ff7f82e9a1ef719a9f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.241ex; height:2.176ex;" alt="{\displaystyle X\times Y.}" loading="lazy"></span> If <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 {\overline {\Gamma (T)}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d8634290b858ef49e47e1b3114bf77ca418952dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.013ex; height:3.676ex;" alt="{\displaystyle {\overline {\Gamma (T)}}}" loading="lazy"></span> is itself the graph of 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 {\overline {T}},}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./aac24a35549af1d05e967145f004e46a79a9a73f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.579ex; height:3.343ex;" alt="{\displaystyle {\overline {T}},}" loading="lazy"></span> <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}">

</mstyle>

</mrow>

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

</semantics>

</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> is called <i>closable</i>, and <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 {\overline {T}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./fd97dce2253b73a45cf6b8dcd84546e3f033aa20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.932ex; height:3.009ex;" alt="{\displaystyle {\overline {T}}}" loading="lazy"></span> is called the <i>closure</i> 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.}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4de28b735beca1303bc5da9ba518a5a22a70a5d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.283ex; height:2.176ex;" alt="{\displaystyle T.}" loading="lazy"></span>

</p><p>So the natural question to ask about linear operators that are not everywhere-defined is whether they are closable. The answer is, "not necessarily"; indeed, every infinite-dimensional normed space admits linear operators that are not closable. As in the case of discontinuous operators considered above, the proof requires the axiom of choice and so is in general nonconstructive, though again, if <i>X</i> is not complete, there are constructible examples.

</p><p>In fact, there is even an example of a linear operator whose graph has closure <i>all</i> 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\times Y.}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./41f39ef78be28dc2b9015ff7f82e9a1ef719a9f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.241ex; height:2.176ex;" alt="{\displaystyle X\times Y.}" loading="lazy"></span> Such an operator is not closable. Let <i>X</i> be the space of <a href="Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial functions</a> from [0,1] 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 \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> and <i>Y</i> the space of polynomial functions from [2,3] 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 \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>. They are subspaces of <i>C</i>([0,1]) and <i>C</i>([2,3]) respectively, and so normed spaces. Define an operator <i>T</i> which takes the polynomial function <i>x</i> ↦ <i>p</i>(<i>x</i>) on [0,1] to the same function on [2,3]. As a consequence of the <a href="Stone%E2%80%93Weierstrass_theorem" title="Stone–Weierstrass theorem">Stone–Weierstrass theorem</a>, the graph of this operator is dense 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\times Y,}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c012108ac139316994be07f2ecf2897382db40f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.241ex; height:2.509ex;" alt="{\displaystyle X\times Y,}" loading="lazy"></span> so this provides a sort of maximally discontinuous linear map (confer <a href="Nowhere_continuous_function" title="Nowhere continuous function">nowhere continuous function</a>). Note that <i>X</i> is not complete here, as must be the case when there is such a constructible map.

</p>

<div class="mw-heading mw-heading2"><h2 id="Impact_for_dual_spaces">Impact for dual spaces</h2></div>

<p>The <a href="Dual_space" title="Dual space">dual space</a> of a topological vector space is the collection of continuous linear maps from the space into the underlying field. Thus the failure of some linear maps to be continuous for infinite-dimensional normed spaces implies that for these spaces, one needs to distinguish the algebraic dual space from the continuous dual space which is then a proper subset. It illustrates the fact that an extra dose of caution is needed in doing analysis on infinite-dimensional spaces as compared to finite-dimensional ones.

</p>

<div class="mw-heading mw-heading2"><h2 id="Beyond_normed_spaces">Beyond normed spaces</h2></div>

<p>The argument for the existence of discontinuous linear maps on normed spaces can be generalized to all metrizable topological vector spaces, especially to all Fréchet spaces, but there exist infinite-dimensional locally convex topological vector spaces such that every functional is continuous.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> On the other hand, the <a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach theorem</a>, which applies to all locally convex spaces, guarantees the existence of many continuous linear functionals, and so a large dual space. In fact, to every convex set, the <a href="Minkowski_gauge" class="mw-redirect" title="Minkowski gauge">Minkowski gauge</a> associates a continuous <a href="Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a>. The upshot is that spaces with fewer convex sets have fewer functionals, and in the worst-case scenario, a space may have no functionals at all other than the zero functional. This is the case for the <a href="Lp_space" title="Lp 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 L^{p}(\mathbb {R} ,dx)}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{p}(\mathbb {R} ,dx)}</annotation>

</semantics>

</math></span><img src="./419c2b508428bb5be5a5a01e7261e04af4f273c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.709ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mathbb {R} ,dx)}" loading="lazy"></span></a> spaces with <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<p<1,}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./73d900254969232ec9def8008e890c03d6d6253b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.338ex; height:2.509ex;" alt="{\displaystyle 0<p<1,}" loading="lazy"></span> from which it follows that these spaces are nonconvex. Note that here is indicated the <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> on the real line. There are other <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^{p}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> spaces with <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<p<1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ea074f5b36db6eff17f1aa84d73e30e3de12c4d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.691ex; height:2.509ex;" alt="{\displaystyle 0<p<1}" loading="lazy"></span> which do have nontrivial dual spaces.

</p><p>Another such example is the space of real-valued <a href="Measurable_function" title="Measurable function">measurable functions</a> on the unit interval with <a href="Quasinorm" title="Quasinorm">quasinorm</a> given by

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|f\|=\int _{I}{\frac {|f(x)|}{1+|f(x)|}}dx.}</annotation>

</semantics>

</math></span></span>

This non-locally convex space has a trivial dual space.

</p><p>One can consider even more general spaces. For example, the existence of a homomorphism between complete separable metric <a href="Group_(mathematics)" title="Group (mathematics)">groups</a> can also be shown nonconstructively.

</p>

<ul><li><a href="Finest_locally_convex_topology" class="mw-redirect" title="Finest locally convex topology">Finest locally convex topology</a> – Vector space with a topology defined by convex open sets<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Sublinear_function" title="Sublinear function">Sublinear function</a> – Type of function in linear algebra</li></ul>

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

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</style><cite id="CITEREFSolovay1970" class="citation cs2"><a href="Robert_M._Solovay" title="Robert M. Solovay">Solovay, Robert M.</a> (1970), "A model of set-theory in which every set of reals is Lebesgue measurable", <i><a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>, Second Series, <b>92</b> (1): <span class="nowrap">1–</span>56, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970696">10.2307/1970696</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970696">1970696</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0265151">0265151</a></cite>.</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFSchechter1996" class="citation cs2">Schechter, Eric (1996), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA136"><i>Handbook of Analysis and Its Foundations</i></a>, Academic Press, p. 136, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780080532998</bdi></cite>.</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">For example, the weak topology w.r.t. the space of all (algebraically) linear functionals.</span>

</li>

</ol></div></div>

<ul><li>Constantin Costara, Dumitru Popa, <i>Exercises in Functional Analysis</i>, Springer, 2003. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-4020-1560-7</bdi>.</li>

<li><a href="Eric_Schechter" title="Eric Schechter">Schechter, Eric</a>, <i>Handbook of Analysis and its Foundations</i>, Academic Press, 1997. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-622760-8</bdi>.</li></ul>

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</style><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="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>

</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>

<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="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>

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

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