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class="comment mw-comment-none">Нет описания правки</span></div><div id="mw-diff-ntitle5"></div><div id="mw-diff-ntitle4"><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&diff=next&oldid=6082941" title="Теорема о замкнутом графике" id="differences-nextlink">Следующая правка →</a></div></td> </tr><tr> <td colspan="2" class="diff-lineno">Строка 1:</td> <td colspan="2" class="diff-lineno">Строка 1:</td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Теорема о замкнутом графике <del class="diffchange diffchange-inline">---</del> важный результат [[Функциональный анализ|функционального анализа]], устанавливающая [[критерий]] [[Ограниченный оператор|ограниченности линейного оператора]] между [[Банахово пространство|банаховыми пространствами]].</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Теорема о замкнутом графике <ins class="diffchange diffchange-inline">&mdash;</ins> важный результат [[Функциональный анализ|функционального анализа]], устанавливающая [[критерий]] [[Ограниченный оператор|ограниченности линейного оператора]] между [[Банахово пространство|банаховыми пространствами]].</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>== Формулировки ==</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>== Формулировки ==</div></td> </tr> </table><hr class='diff-hr' id='mw-oldid' /> <h2 class='diff-currentversion-title'>Версия от 20:04, 27 октября 2007</h2> <div class="mw-content-ltr mw-parser-output" lang="ru" dir="ltr"><p>Теорема о замкнутом графике — 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>, устанавливающая <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 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></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 gf(T):=\{(x,y)\in X\times Y|y=Tx\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>X</mi> <mo>×</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo>=</mo> <mi>T</mi> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle gf(T):=\{(x,y)\in X\times Y|y=Tx\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd7324126217425f4efc241aa9589ca90ac35f84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.54ex; height:2.843ex;" alt="{\displaystyle gf(T):=\{(x,y)\in X\times Y|y=Tx\}}"></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>.</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> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Обобщения"><span id=".D0.9E.D0.B1.D0.BE.D0.B1.D1.89.D0.B5.D0.BD.D0.B8.D1.8F"></span>Обобщения</h2></div> <p>Теорема о замкнутом графике в первой из приведённых формулировок сохраняет силу и при некотором ослаблении требований; именно, достаточно потребовать, чтобы <i>X</i> было <a href="/w/index.php?title=%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%BE%D0%B2%D0%B5%D1%88%D0%B5%D0%BD%D0%BD%D0%BE%D0%B5_%D0%BB%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%BE%D0%B5_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%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&action=edit&redlink=1" class="new" title="Уравновешенное линейное топологическое пространство (страница отсутствует)">уравновешенным 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>. </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></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 (англ.)</a></li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Источник — <a dir="ltr" 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data-title="Sætningen om lukkede grafer" data-language-autonym="Dansk" data-language-local-name="датский" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Satz_vom_abgeschlossenen_Graphen" title="Satz vom abgeschlossenen Graphen — немецкий" 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 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data-language-local-name="итальянский" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%96%89%E3%82%B0%E3%83%A9%E3%83%95%E5%AE%9A%E7%90%86" title="閉グラフ定理 — японский" lang="ja" hreflang="ja" data-title="閉グラフ定理" data-language-autonym="日本語" data-language-local-name="японский" 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 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