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class="noprint"></div> <div id="mw-head-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="siteNotice"></div> <div class="mw-indicators"> </div> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Теорема о замкнутом графике</span></h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">Материал из Википедии — свободной энциклопедии</div> <div id="contentSub"><div id="mw-content-subtitle"><div id="mw-fr-revision-messages"><div class="cdx-message mw-fr-message-box cdx-message--block cdx-message--notice mw-fr-basic mw-fr-draft-not-synced plainlinks noprint"><span class="cdx-message__icon"></span><div class="cdx-message__content">Текущая версия страницы пока <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%9F%D1%80%D0%BE%D0%B2%D0%B5%D1%80%D0%BA%D0%B0_%D1%81%D1%82%D0%B0%D1%82%D0%B5%D0%B9/%D0%9F%D0%BE%D1%8F%D1%81%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B4%D0%BB%D1%8F_%D1%87%D0%B8%D1%82%D0%B0%D1%82%D0%B5%D0%BB%D0%B5%D0%B9" title="Википедия:Проверка статей/Пояснение для читателей">не проверялась</a> опытными участниками и может значительно отличаться от <a class="external text" href="https://ru.wikipedia.org/w/index.php?title=%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BE_%D0%B7%D0%B0%D0%BC%D0%BA%D0%BD%D1%83%D1%82%D0%BE%D0%BC_%D0%B3%D1%80%D0%B0%D1%84%D0%B8%D0%BA%D0%B5&stable=1">версии, проверенной 16 мая 2020 года</a>; проверки требует <a class="external text" href="https://ru.wikipedia.org/w/index.php?title=%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BE_%D0%B7%D0%B0%D0%BC%D0%BA%D0%BD%D1%83%D1%82%D0%BE%D0%BC_%D0%B3%D1%80%D0%B0%D1%84%D0%B8%D0%BA%D0%B5&oldid=107077979&diff=cur&diffonly=0">1 правка</a>.</div></div></div></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Перейти к навигации</a> <a class="mw-jump-link" href="#searchInput">Перейти к поиску</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ru" dir="ltr"><p><b>Теорема о замкнутом графике</b> — важный результат <a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Функциональный анализ">функционального анализа</a>, устанавливающий <a href="/wiki/%D0%9A%D1%80%D0%B8%D1%82%D0%B5%D1%80%D0%B8%D0%B9" title="Критерий">критерий</a> <a href="/wiki/%D0%9E%D0%B3%D1%80%D0%B0%D0%BD%D0%B8%D1%87%D0%B5%D0%BD%D0%BD%D1%8B%D0%B9_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D0%BE%D1%80" title="Ограниченный оператор">ограниченности линейного оператора</a> между <a href="/wiki/%D0%91%D0%B0%D0%BD%D0%B0%D1%85%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Банахово пространство">банаховыми пространствами</a>. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="ru" dir="ltr"><h2 id="mw-toc-heading">Содержание</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Формулировки"><span class="tocnumber">1</span> <span class="toctext">Формулировки</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Замечания"><span class="tocnumber">2</span> <span class="toctext">Замечания</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Следствия"><span class="tocnumber">3</span> <span class="toctext">Следствия</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Ссылки"><span class="tocnumber">4</span> <span class="toctext">Ссылки</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Формулировки"><span id=".D0.A4.D0.BE.D1.80.D0.BC.D1.83.D0.BB.D0.B8.D1.80.D0.BE.D0.B2.D0.BA.D0.B8"></span>Формулировки</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BE_%D0%B7%D0%B0%D0%BC%D0%BA%D0%BD%D1%83%D1%82%D0%BE%D0%BC_%D0%B3%D1%80%D0%B0%D1%84%D0%B8%D0%BA%D0%B5&veaction=edit&section=1" title="Редактировать раздел «Формулировки»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BE_%D0%B7%D0%B0%D0%BC%D0%BA%D0%BD%D1%83%D1%82%D0%BE%D0%BC_%D0%B3%D1%80%D0%B0%D1%84%D0%B8%D0%BA%D0%B5&action=edit&section=1" title="Редактировать код раздела «Формулировки»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D1%8B%D0%B9_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D0%BE%D1%80" class="mw-redirect" title="Линейный оператор">Линейный оператор</a> <span class="mwe-math-element"><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:\,X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mspace width="thinmathspace" /> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:\,X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/401a1cacdda670f7d0c1212f8cb2a346204593da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.328ex; height:2.176ex;" alt="{\displaystyle T:\,X\to Y}"></span> между <a href="/wiki/%D0%91%D0%B0%D0%BD%D0%B0%D1%85%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Банахово пространство">банаховыми пространствами</a> <i>X</i> и <i>Y</i> ограничен тогда и только тогда, когда его график <span class="mwe-math-element"><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}:=\{(x,Tx)\in X\times Y:x\in D(T)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>T</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>X</mi> <mo>×</mo> <mi>Y</mi> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{T}:=\{(x,Tx)\in X\times Y:x\in D(T)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/355af9b4f130ff1912252856f1224d0249e7b52e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.963ex; height:2.843ex;" alt="{\displaystyle \Gamma _{T}:=\{(x,Tx)\in X\times Y:x\in D(T)\}}"></span> <a href="/wiki/%D0%97%D0%B0%D0%BC%D0%BA%D0%BD%D1%83%D1%82%D0%BE%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Замкнутое множество">замкнут</a> в пространстве <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1613c1ff4b6fbfb6c80a8da83e90ad28f0ab3483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.594ex; height:2.176ex;" alt="{\displaystyle X\times Y}"></span> и оператор определен на всём пространстве <i>X</i> (т.е. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(T)=X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(T)=X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2bd0f7e64e6f741a9c1f210c60ec982709574ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.448ex; height:2.843ex;" alt="{\displaystyle D(T)=X}"></span>).</li> <li><a href="/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D1%8B%D0%B9_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D0%BE%D1%80" class="mw-redirect" title="Линейный оператор">Линейный оператор</a> <span class="mwe-math-element"><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:\,X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mspace width="thinmathspace" /> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:\,X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/401a1cacdda670f7d0c1212f8cb2a346204593da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.328ex; height:2.176ex;" alt="{\displaystyle T:\,X\to Y}"></span> между <a href="/wiki/%D0%91%D0%B0%D0%BD%D0%B0%D1%85%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Банахово пространство">банаховыми пространствами</a> <i>X</i> и <i>Y</i> ограничен тогда и только тогда, когда для любой последовательности <span class="mwe-math-element"><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_{n}\}_{n\in \mathbb {N} }\subset X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo>⊂</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x_{n}\}_{n\in \mathbb {N} }\subset X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98ee3339ff4106339359dec46a88dacb6b6f52ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.453ex; height:2.843ex;" alt="{\displaystyle \{x_{n}\}_{n\in \mathbb {N} }\subset X}"></span>, такой что <span class="mwe-math-element"><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_{n}\to x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}\to x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd1fbf73434aa72a428b9b100919b1f6e26f6c84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.492ex; height:2.176ex;" alt="{\displaystyle x_{n}\to x}"></span> и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Tx_{n}\to y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Tx_{n}\to y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbf5c651b18b79723b61770d138c95eae8f0f81b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.954ex; height:2.509ex;" alt="{\displaystyle Tx_{n}\to y}"></span>, выполняется <span class="mwe-math-element"><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=Tx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>T</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=Tx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30f0d5945d56698798223d14135395194407274f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.22ex; height:2.509ex;" alt="{\displaystyle y=Tx}"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Замечания"><span id=".D0.97.D0.B0.D0.BC.D0.B5.D1.87.D0.B0.D0.BD.D0.B8.D1.8F"></span>Замечания</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BE_%D0%B7%D0%B0%D0%BC%D0%BA%D0%BD%D1%83%D1%82%D0%BE%D0%BC_%D0%B3%D1%80%D0%B0%D1%84%D0%B8%D0%BA%D0%B5&veaction=edit&section=2" title="Редактировать раздел «Замечания»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BE_%D0%B7%D0%B0%D0%BC%D0%BA%D0%BD%D1%83%D1%82%D0%BE%D0%BC_%D0%B3%D1%80%D0%B0%D1%84%D0%B8%D0%BA%D0%B5&action=edit&section=2" title="Редактировать код раздела «Замечания»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Первая из приведённых формулировок сохраняет силу и при некотором ослаблении требований; а именно, достаточно потребовать, чтобы <i>X</i> было <a href="/wiki/%D0%91%D0%BE%D1%87%D0%B5%D1%87%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Бочечное пространство">бочечным линейным топологическим пространством</a>, а <i>Y</i> — <a href="/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%A4%D1%80%D0%B5%D1%88%D0%B5" title="Пространство Фреше">пространством Фреше</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Следствия"><span id=".D0.A1.D0.BB.D0.B5.D0.B4.D1.81.D1.82.D0.B2.D0.B8.D1.8F"></span>Следствия</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BE_%D0%B7%D0%B0%D0%BC%D0%BA%D0%BD%D1%83%D1%82%D0%BE%D0%BC_%D0%B3%D1%80%D0%B0%D1%84%D0%B8%D0%BA%D0%B5&veaction=edit&section=3" title="Редактировать раздел «Следствия»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BE_%D0%B7%D0%B0%D0%BC%D0%BA%D0%BD%D1%83%D1%82%D0%BE%D0%BC_%D0%B3%D1%80%D0%B0%D1%84%D0%B8%D0%BA%D0%B5&action=edit&section=3" title="Редактировать код раздела «Следствия»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Из теоремы о замкнутом графике следует <a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%A5%D0%B5%D0%BB%D0%BB%D0%B8%D0%BD%D0%B3%D0%B5%D1%80%D0%B0_%E2%80%94_%D0%A2%D1%91%D0%BF%D0%BB%D0%B8%D1%86%D0%B0" title="Теорема Хеллингера — Тёплица">теорема Хеллингера — Тёплица</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Ссылки"><span id=".D0.A1.D1.81.D1.8B.D0.BB.D0.BA.D0.B8"></span>Ссылки</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BE_%D0%B7%D0%B0%D0%BC%D0%BA%D0%BD%D1%83%D1%82%D0%BE%D0%BC_%D0%B3%D1%80%D0%B0%D1%84%D0%B8%D0%BA%D0%B5&veaction=edit&section=4" title="Редактировать раздел «Ссылки»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BE_%D0%B7%D0%B0%D0%BC%D0%BA%D0%BD%D1%83%D1%82%D0%BE%D0%BC_%D0%B3%D1%80%D0%B0%D1%84%D0%B8%D0%BA%D0%B5&action=edit&section=4" title="Редактировать код раздела «Ссылки»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://planetmath.org/?op=getobj&from=objects&id=6472%7C">Доказательство теоремы о замкнутом графике</a> <small class="ref-info" style="cursor:help;" title="на английском языке">(англ.)</small></li></ul> <style data-mw-deduplicate="TemplateStyles:r137874932">.mw-parser-output .ambox{border:1px solid var(--border-color-base,#a2a9b1);border-left:10px solid #36c;background:var(--background-color-neutral-subtle,#f8f9fa);box-sizing:border-box;margin:0 10%}html body.mediawiki.skin-minerva .mw-parser-output .ambox{border-width:0 0 0 4px}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output 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0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}.mw-parser-output .ambox .mbox-textsmall-div{font-size:90%}html.client-js body.skin-minerva .mw-parser-output .mbox-text-span{margin-left:23px!important}@media(max-width:1366px){.mw-parser-output .ambox{margin-left:6%;margin-right:6%}}@media(max-width:719px){.mw-parser-output .ambox{margin-left:0;margin-right:0}}</style><table class="plainlinks metadata ambox ambox-style" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%9A%D0%B0%D0%BA_%D0%BF%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D1%8C_%D1%81%D1%82%D0%B0%D1%82%D1%8C%D0%B8" title="Улучшение статьи"><img alt="Улучшение статьи" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/40px-Wiki_letter_w.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/60px-Wiki_letter_w.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/80px-Wiki_letter_w.svg.png 2x" data-file-width="44" data-file-height="44" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-div"><b>Для улучшения этой статьи по математике <a href="/wiki/%D0%A8%D0%B0%D0%B1%D0%BB%D0%BE%D0%BD:Rq/%D0%9F%D0%BE%D1%8F%D1%81%D0%BD%D0%B5%D0%BD%D0%B8%D1%8F" title="Шаблон:Rq/Пояснения">желательно</a>:</b><ul style="margin-top: 0;"> <li><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%9F%D0%BE%D0%B8%D1%81%D0%BA_%D0%B8%D1%81%D1%82%D0%BE%D1%87%D0%BD%D0%B8%D0%BA%D0%BE%D0%B2" title="Википедия:Поиск источников">Найти</a> и оформить в виде <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BD%D0%BE%D1%81%D0%BA%D0%B8" title="Википедия:Сноски">сносок</a> ссылки на независимые <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%90%D0%B2%D1%82%D0%BE%D1%80%D0%B8%D1%82%D0%B5%D1%82%D0%BD%D1%8B%D0%B5_%D0%B8%D1%81%D1%82%D0%BE%D1%87%D0%BD%D0%B8%D0%BA%D0%B8" title="Википедия:Авторитетные источники">авторитетные источники</a>, <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%9F%D1%80%D0%BE%D0%B2%D0%B5%D1%80%D1%8F%D0%B5%D0%BC%D0%BE%D1%81%D1%82%D1%8C" title="Википедия:Проверяемость">подтверждающие написанное</a>.</li><li>Дополнить статью (статья <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A0%D0%B0%D0%B7%D0%BC%D0%B5%D1%80_%D1%81%D1%82%D0%B0%D1%82%D0%B5%D0%B9#Общие_рекомендации" title="Википедия:Размер статей">слишком короткая</a> либо содержит лишь <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BB%D0%BE%D0%B2%D0%B0%D1%80%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Википедия:Словарность">словарное определение</a>).</li></ul></div><div class="mbox-textsmall-div hide-when-compact">После исправления проблемы исключите её из списка. Удалите шаблон, если устранены все недостатки.</div></td></tr></tbody></table>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Источник — <a dir="ltr" href="https://ru.wikipedia.org/w/index.php?title=Теорема_о_замкнутом_графике&oldid=127335204">https://ru.wikipedia.org/w/index.php?title=Теорема_о_замкнутом_графике&oldid=127335204</a></div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D0%B8" title="Служебная:Категории">Категории</a>: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D1%8B_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B3%D0%BE_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0" title="Категория:Теоремы функционального анализа">Теоремы функционального анализа</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Категория:Функциональный анализ">Функциональный анализ</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D0%BE%D1%80%D0%BE%D0%B2" title="Категория:Теория операторов">Теория операторов</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Скрытые категории: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D1%82%D0%B0%D1%82%D1%8C%D0%B8_%D0%B1%D0%B5%D0%B7_%D1%81%D1%81%D1%8B%D0%BB%D0%BE%D0%BA_%D0%BD%D0%B0_%D0%B8%D1%81%D1%82%D0%BE%D1%87%D0%BD%D0%B8%D0%BA%D0%B8" title="Категория:Википедия:Статьи без ссылок на источники">Википедия:Статьи без ссылок на источники</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D1%82%D0%B0%D1%82%D1%8C%D0%B8_%D0%B1%D0%B5%D0%B7_%D0%B8%D1%81%D1%82%D0%BE%D1%87%D0%BD%D0%B8%D0%BA%D0%BE%D0%B2_(%D1%82%D0%B8%D0%BF:_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0)" title="Категория:Википедия:Статьи без источников (тип: теорема)">Википедия:Статьи без источников (тип: теорема)</a></li><li><a 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href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D1%82%D0%B0%D1%82%D1%8C%D0%B8_%D1%81_%D1%88%D0%B0%D0%B1%D0%BB%D0%BE%D0%BD%D0%B0%D0%BC%D0%B8_%D0%BD%D0%B5%D0%B4%D0%BE%D1%81%D1%82%D0%B0%D1%82%D0%BA%D0%BE%D0%B2_%D0%BF%D0%BE_%D0%B0%D0%BB%D1%84%D0%B0%D0%B2%D0%B8%D1%82%D1%83" title="Категория:Википедия:Статьи с шаблонами недостатков по алфавиту">Википедия:Статьи с шаблонами недостатков по алфавиту</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Навигация</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Персональные инструменты</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Страница участника для 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lang="de" hreflang="de" data-title="Satz vom abgeschlossenen Graphen" data-language-autonym="Deutsch" data-language-local-name="немецкий" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Closed_graph_theorem" title="Closed graph theorem — английский" lang="en" hreflang="en" data-title="Closed graph theorem" data-language-autonym="English" data-language-local-name="английский" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_de_la_gr%C3%A1fica_cerrada" title="Teorema de la gráfica cerrada — испанский" lang="es" hreflang="es" data-title="Teorema de la gráfica cerrada" data-language-autonym="Español" data-language-local-name="испанский" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a 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class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%AB%ED%9E%8C_%EA%B7%B8%EB%9E%98%ED%94%84_%EC%A0%95%EB%A6%AC" title="닫힌 그래프 정리 — корейский" lang="ko" hreflang="ko" data-title="닫힌 그래프 정리" data-language-autonym="한국어" data-language-local-name="корейский" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_van_de_gesloten_grafiek" title="Stelling van de gesloten grafiek — нидерландский" lang="nl" hreflang="nl" data-title="Stelling van de gesloten grafiek" data-language-autonym="Nederlands" data-language-local-name="нидерландский" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie_o_wykresie_domkni%C4%99tym" title="Twierdzenie o wykresie domkniętym — польский" lang="pl" hreflang="pl" data-title="Twierdzenie o wykresie domkniętym" data-language-autonym="Polski" data-language-local-name="польский" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Teorema_d%C3%ABl_gr%C3%A0fich_sar%C3%A0" title="Teorema dël gràfich sarà — Piedmontese" lang="pms" hreflang="pms" data-title="Teorema dël gràfich sarà" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_do_gr%C3%A1fico_fechado" title="Teorema do gráfico fechado — португальский" lang="pt" hreflang="pt" data-title="Teorema do gráfico fechado" data-language-autonym="Português" data-language-local-name="португальский" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BE_%D0%B7%D0%B0%D1%82%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BE%D0%BC_%D0%B3%D1%80%D0%B0%D1%84%D0%B8%D0%BA%D1%83" title="Теорема о затвореном графику — сербский" lang="sr" hreflang="sr" data-title="Теорема о затвореном графику" data-language-autonym="Српски / srpski" data-language-local-name="сербский" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Satsen_om_den_slutna_grafen" title="Satsen om den slutna grafen — шведский" lang="sv" hreflang="sv" data-title="Satsen om den slutna grafen" data-language-autonym="Svenska" data-language-local-name="шведский" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%91%D0%B0%D0%BD%D0%B0%D1%85%D0%B0_%D0%BF%D1%80%D0%BE_%D0%B7%D0%B0%D0%BC%D0%BA%D0%BD%D0%B5%D0%BD%D0%B8%D0%B9_%D0%B3%D1%80%D0%B0%D1%84%D1%96%D0%BA" title="Теорема Банаха про замкнений графік — украинский" lang="uk" hreflang="uk" data-title="Теорема Банаха про замкнений графік" data-language-autonym="Українська" data-language-local-name="украинский" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%96%89%E5%9C%96%E5%83%8F%E5%AE%9A%E7%90%86" title="閉圖像定理 — китайский" lang="zh" hreflang="zh" data-title="閉圖像定理" data-language-autonym="中文" data-language-local-name="китайский" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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