Uniform boundedness principle

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Available in 20 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-20" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">20 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_de_Banach-Steinhaus" title="Teorema de Banach-Steinhaus – Catalan" lang="ca" hreflang="ca" data-title="Teorema de Banach-Steinhaus" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Banachova%E2%80%93Steinhausova_v%C4%9Bta" title="Banachova–Steinhausova věta – Czech" lang="cs" hreflang="cs" data-title="Banachova–Steinhausova věta" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Satz_von_Banach-Steinhaus" title="Satz von Banach-Steinhaus – German" lang="de" hreflang="de" data-title="Satz von Banach-Steinhaus" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Principio_de_acotaci%C3%B3n_uniforme" title="Principio de acotación uniforme – Spanish" lang="es" hreflang="es" data-title="Principio de acotación uniforme" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B5%D9%84_%DA%A9%D8%B1%D8%A7%D9%86%E2%80%8C%D8%AF%D8%A7%D8%B1%DB%8C_%DB%8C%DA%A9%D9%86%D9%88%D8%A7%D8%AE%D8%AA" title="اصل کران‌داری یکنواخت – Persian" lang="fa" hreflang="fa" data-title="اصل کران‌داری یکنواخت" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Banach-Steinhaus" title="Théorème de Banach-Steinhaus – French" lang="fr" hreflang="fr" data-title="Théorème de Banach-Steinhaus" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B7%A0%EB%93%B1_%EC%9C%A0%EA%B3%84%EC%84%B1_%EC%9B%90%EB%A6%AC" title="균등 유계성 원리 – Korean" lang="ko" hreflang="ko" data-title="균등 유계성 원리" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Principio_dell%27uniforme_limitatezza" title="Principio dell'uniforme limitatezza – Italian" lang="it" hreflang="it" data-title="Principio dell'uniforme limitatezza" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%91%D7%A0%D7%9A-%D7%A9%D7%98%D7%99%D7%99%D7%A0%D7%94%D7%90%D7%95%D7%A1" title="משפט בנך-שטיינהאוס – Hebrew" lang="he" hreflang="he" data-title="משפט בנך-שטיינהאוס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Principe_van_uniforme_begrensdheid" title="Principe van uniforme begrensdheid – Dutch" lang="nl" hreflang="nl" data-title="Principe van uniforme begrensdheid" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Teorema_%C3%ABd_Banach-Steinhaus" title="Teorema ëd Banach-Steinhaus – Piedmontese" lang="pms" hreflang="pms" data-title="Teorema ëd Banach-Steinhaus" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie_Banacha-Steinhausa" title="Twierdzenie Banacha-Steinhausa – Polish" lang="pl" hreflang="pl" data-title="Twierdzenie Banacha-Steinhausa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_de_Banach-Steinhaus" title="Teorema de Banach-Steinhaus – Portuguese" lang="pt" hreflang="pt" data-title="Teorema de Banach-Steinhaus" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D0%BD%D1%86%D0%B8%D0%BF_%D1%80%D0%B0%D0%B2%D0%BD%D0%BE%D0%BC%D0%B5%D1%80%D0%BD%D0%BE%D0%B9_%D0%BE%D0%B3%D1%80%D0%B0%D0%BD%D0%B8%D1%87%D0%B5%D0%BD%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Принцип равномерной ограниченности – Russian" lang="ru" hreflang="ru" data-title="Принцип равномерной ограниченности" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Tiurema_di_Banach-Steinhaus" title="Tiurema di Banach-Steinhaus – Sicilian" lang="scn" hreflang="scn" data-title="Tiurema di Banach-Steinhaus" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</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%91%D0%B0%D0%BD%D0%B0%D1%85%D0%B0-%D0%A8%D1%82%D0%B0%D1%98%D0%BD%D1%85%D0%B0%D1%83%D1%81%D0%B0" title="Теорема Банаха-Штајнхауса – Serbian" lang="sr" hreflang="sr" data-title="Теорема Банаха-Штајнхауса" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" 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/Banach-Steinhaus_sats" title="Banach-Steinhaus sats – Swedish" lang="sv" hreflang="sv" data-title="Banach-Steinhaus sats" data-language-autonym="Svenska" data-language-local-name="Swedish" 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_%E2%80%94_%D0%A8%D1%82%D0%B5%D0%B9%D0%BD%D0%B3%D0%B0%D1%83%D0%B7%D0%B0" title="Теорема Банаха — Штейнгауза – Ukrainian" lang="uk" hreflang="uk" data-title="Теорема Банаха — Штейнгауза" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_Banach-Steinhause" title="Định lý Banach-Steinhause – Vietnamese" lang="vi" hreflang="vi" data-title="Định lý Banach-Steinhause" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%80%E8%87%B4%E6%9C%89%E7%95%8C%E6%80%A7%E5%8E%9F%E7%90%86" title="一致有界性原理 – Chinese" lang="zh" hreflang="zh" data-title="一致有界性原理" data-language-autonym="中文" data-language-local-name="Chinese" 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 href="https://www.wikidata.org/wiki/Special:EntityPage/Q1426292#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> 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data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the definition of uniformly bounded functions, see <a href="/wiki/Uniform_boundedness" title="Uniform boundedness">Uniform boundedness</a>. For the conjectures in number theory and algebraic geometry, see <a href="/wiki/Uniform_boundedness_conjecture_(disambiguation)" class="mw-redirect mw-disambig" title="Uniform boundedness conjecture (disambiguation)">Uniform boundedness conjecture (disambiguation)</a>.</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>uniform boundedness principle</b> or <b>Banach–Steinhaus theorem</b> is one of the fundamental results in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>. Together with the <a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach theorem</a> and the <a href="/wiki/Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">open mapping theorem</a>, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of <a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">continuous linear operators</a> (and thus <a href="/wiki/Bounded_operator" title="Bounded operator">bounded operators</a>) whose domain is a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>, pointwise <a href="/wiki/Bounded_set" title="Bounded set">boundedness</a> is equivalent to uniform boundedness in <a href="/wiki/Operator_norm" title="Operator norm">operator norm</a>. </p><p>The theorem was first published in 1927 by <a href="/wiki/Stefan_Banach" title="Stefan Banach">Stefan Banach</a> and <a href="/wiki/Hugo_Steinhaus" title="Hugo Steinhaus">Hugo Steinhaus</a>, but it was also proven independently by <a href="/wiki/Hans_Hahn_(mathematician)" title="Hans Hahn (mathematician)">Hans Hahn</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Theorem">Theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_boundedness_principle&action=edit&section=1" title="Edit section: Theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name">Uniform Boundedness Principle</strong><span class="theoreme-tiret"> — </span>Let <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> be a <a href="/wiki/Banach_space" title="Banach space">Banach space</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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> a <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a> and <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 B(X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/161c8cf2a741eaf454166d4c6bf263ff1420d457" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.361ex; height:2.843ex;" alt="{\displaystyle B(X,Y)}" /></span> the space of all <a href="/wiki/Continuous_linear_operators" class="mw-redirect" title="Continuous linear operators">continuous linear operators</a> from <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> into <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span>. Suppose that <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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" /></span> is a collection of continuous linear operators from <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> to <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" /></span> If, for every <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\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{T\in F}\|T(x)\|_{Y}<\infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>∈</mo> <mi>F</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{T\in F}\|T(x)\|_{Y}<\infty ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fed32dea213b322e6a6a595ec9a4c561b3caab7b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.544ex; height:4.509ex;" alt="{\displaystyle \sup _{T\in F}\|T(x)\|_{Y}<\infty ,}" /></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{T\in F}\|T\|_{B(X,Y)}<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>∈</mo> <mi>F</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{T\in F}\|T\|_{B(X,Y)}<\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3994c0a5213341963400a1556d31f2c33df0797c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.789ex; height:4.509ex;" alt="{\displaystyle \sup _{T\in F}\|T\|_{B(X,Y)}<\infty .}" /></span> </p> </div> <p>The first inequality (that is, <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="{\textstyle \sup _{T\in F}\|T(x)\|<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>∈</mo> <mi>F</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sup _{T\in F}\|T(x)\|<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3cef68865fdf25b2bed774efdd053a46fc90ccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.127ex; height:2.843ex;" alt="{\textstyle \sup _{T\in F}\|T(x)\|<\infty }" /></span> for all <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}" /></span>) states that the functionals in <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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" /></span> are pointwise bounded while the second states that they are uniformly bounded. The second supremum always equals <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{T\in F}\|T\|_{B(X,Y)}=\sup _{\stackrel {T\in F,}{\|x\|\leq 1}}\|T(x)\|_{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>∈</mo> <mi>F</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>∈</mo> <mi>F</mi> <mo>,</mo> </mrow> </mover> </mrow> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{T\in F}\|T\|_{B(X,Y)}=\sup _{\stackrel {T\in F,}{\|x\|\leq 1}}\|T(x)\|_{Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83bda74296a2a83d246c5b216ff05b546d3a5969" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:30.477ex; height:6.509ex;" alt="{\displaystyle \sup _{T\in F}\|T\|_{B(X,Y)}=\sup _{\stackrel {T\in F,}{\|x\|\leq 1}}\|T(x)\|_{Y}}" /></span> and if <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> is not the trivial vector space (or if the supremum is taken over <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 [0,\infty ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\infty ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52088d5605716e18068a460dec118214954a68e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.814ex; height:2.843ex;" alt="{\displaystyle [0,\infty ]}" /></span> rather than <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 [-\infty ,\infty ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-\infty ,\infty ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13233867b861889693a36843d98e51d90d38f9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.783ex; height:2.843ex;" alt="{\displaystyle [-\infty ,\infty ]}" /></span>) then closed unit ball can be replaced with the unit sphere <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{T\in F}\|T\|_{B(X,Y)}=\sup _{\stackrel {T\in F,}{\|x\|=1}}\|T(x)\|_{Y}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>∈</mo> <mi>F</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>∈</mo> <mi>F</mi> <mo>,</mo> </mrow> </mover> </mrow> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{T\in F}\|T\|_{B(X,Y)}=\sup _{\stackrel {T\in F,}{\|x\|=1}}\|T(x)\|_{Y}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25a0deb3939ff34c7301a5d4e6006763e4fc7383" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:31.123ex; height:6.509ex;" alt="{\displaystyle \sup _{T\in F}\|T\|_{B(X,Y)}=\sup _{\stackrel {T\in F,}{\|x\|=1}}\|T(x)\|_{Y}.}" /></span> </p><p>The completeness of the Banach space <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> enables the following short proof, using the <a href="/wiki/Baire_category_theorem" title="Baire category theorem">Baire category theorem</a>. </p> <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style=""><strong>Proof</strong> <p>Suppose <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> is a Banach space and that for every <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\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" /></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{T\in F}\|T(x)\|_{Y}<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>∈</mo> <mi>F</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{T\in F}\|T(x)\|_{Y}<\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2c24783fcf959c11e8b9748062838da1b5df15f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.544ex; height:4.509ex;" alt="{\displaystyle \sup _{T\in F}\|T(x)\|_{Y}<\infty .}" /></span> </p><p>For every integer <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 n\in \mathbb {N} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b98e213fe7ef48da0be47453bc1bb66f37f4eec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.56ex; height:2.509ex;" alt="{\displaystyle n\in \mathbb {N} ,}" /></span> let <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n}=\left\{x\in X\ :\ \sup _{T\in F}\|T(x)\|_{Y}\leq n\right\}.}"> <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>=</mo> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>∈</mo> <mi>F</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>≤</mo> <mi>n</mi> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}=\left\{x\in X\ :\ \sup _{T\in F}\|T(x)\|_{Y}\leq n\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e8c2483751431539c47da626bc9f2afa65adcf1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.365ex; height:6.176ex;" alt="{\displaystyle X_{n}=\left\{x\in X\ :\ \sup _{T\in F}\|T(x)\|_{Y}\leq n\right\}.}" /></span> </p><p>Each set <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}" /></span> is a <a href="/wiki/Closed_set" title="Closed set">closed set</a> and by the assumption, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcup _{n\in \mathbb {N} }X_{n}=X\neq \varnothing .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>X</mi> <mo>≠</mo> <mi class="MJX-variant">∅</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup _{n\in \mathbb {N} }X_{n}=X\neq \varnothing .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06ace97a13f52679adc8cb5c41a7f3730597551c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.431ex; height:5.676ex;" alt="{\displaystyle \bigcup _{n\in \mathbb {N} }X_{n}=X\neq \varnothing .}" /></span> </p><p>By the <a href="/wiki/Baire_category_theorem" title="Baire category theorem">Baire category theorem</a> for the non-empty <a href="/wiki/Complete_metric_space" title="Complete metric space">complete metric space</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 X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" /></span> there exists some <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 m\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42411e85d874a733209223302bbd8d5e3ad04cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {N} }" /></span> such that <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_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb527ad02937e44b7e1e06f84afe6ca5b4875606" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.599ex; height:2.509ex;" alt="{\displaystyle X_{m}}" /></span> has non-empty <a href="/wiki/Interior_(topology)" title="Interior (topology)">interior</a>; that is, there exist <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_{0}\in X_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\in X_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b99d619cc969640204af800ced87ef8441a6f8e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.824ex; height:2.509ex;" alt="{\displaystyle x_{0}\in X_{m}}" /></span> and <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 \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" /></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {B_{\varepsilon }(x_{0})}}~:=~\left\{x\in X\,:\,\|x-x_{0}\|\leq \varepsilon \right\}~\subseteq ~X_{m}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mtext> </mtext> <mo>:=</mo> <mtext> </mtext> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mspace width="thinmathspace"></mspace> <mo>:</mo> <mspace width="thinmathspace"></mspace> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mi>ε</mi> </mrow> <mo>}</mo> </mrow> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {B_{\varepsilon }(x_{0})}}~:=~\left\{x\in X\,:\,\|x-x_{0}\|\leq \varepsilon \right\}~\subseteq ~X_{m}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4bb0950788481fc4012987697c845d733b4d62" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.505ex; height:3.676ex;" alt="{\displaystyle {\overline {B_{\varepsilon }(x_{0})}}~:=~\left\{x\in X\,:\,\|x-x_{0}\|\leq \varepsilon \right\}~\subseteq ~X_{m}.}" /></span> </p><p>Let <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 u\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c358e942a980691c36011ffbfd456695180544e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle u\in X}" /></span> with <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 \|u\|\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|u\|\leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d529b8eb5c62d431e80e0981cf83b381d61384f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.916ex; height:2.843ex;" alt="{\displaystyle \|u\|\leq 1}" /></span> and <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\in F.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>∈</mo> <mi>F</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\in F.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4c6d61bb2eab36fa01bb1821e40bc4b6d1ab8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.865ex; height:2.176ex;" alt="{\displaystyle T\in F.}" /></span> Then: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\|T(u)\|_{Y}&=\varepsilon ^{-1}\left\|T\left(x_{0}+\varepsilon u\right)-T\left(x_{0}\right)\right\|_{Y}&[{\text{by linearity of }}T]\\&\leq \varepsilon ^{-1}\left(\left\|T(x_{0}+\varepsilon u)\right\|_{Y}+\left\|T(x_{0})\right\|_{Y}\right)\\&\leq \varepsilon ^{-1}(m+m).&[{\text{since }}\ x_{0}+\varepsilon u,\ x_{0}\in X_{m}]\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>T</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mi>u</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>T</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>by linearity of </mtext> </mrow> <mi>T</mi> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>≤</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>≤</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> <mtd> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>since </mtext> </mrow> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mi>u</mi> <mo>,</mo> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\|T(u)\|_{Y}&=\varepsilon ^{-1}\left\|T\left(x_{0}+\varepsilon u\right)-T\left(x_{0}\right)\right\|_{Y}&[{\text{by linearity of }}T]\\&\leq \varepsilon ^{-1}\left(\left\|T(x_{0}+\varepsilon u)\right\|_{Y}+\left\|T(x_{0})\right\|_{Y}\right)\\&\leq \varepsilon ^{-1}(m+m).&[{\text{since }}\ x_{0}+\varepsilon u,\ x_{0}\in X_{m}]\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/586805bd601cab526a89e01c79fd8eb5b805a3ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.154ex; margin-bottom: -0.184ex; width:75.521ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}\|T(u)\|_{Y}&=\varepsilon ^{-1}\left\|T\left(x_{0}+\varepsilon u\right)-T\left(x_{0}\right)\right\|_{Y}&[{\text{by linearity of }}T]\\&\leq \varepsilon ^{-1}\left(\left\|T(x_{0}+\varepsilon u)\right\|_{Y}+\left\|T(x_{0})\right\|_{Y}\right)\\&\leq \varepsilon ^{-1}(m+m).&[{\text{since }}\ x_{0}+\varepsilon u,\ x_{0}\in X_{m}]\\\end{aligned}}}" /></span> </p><p>Taking the supremum over <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 u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" /></span> in the unit ball of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> and over <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\in F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>∈</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\in F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e35be958eeca0a6fea8035267a4dc72b00d2248d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.218ex; height:2.176ex;" alt="{\displaystyle T\in F}" /></span> it follows that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{T\in F}\|T\|_{B(X,Y)}~\leq ~2\varepsilon ^{-1}m~<~\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>∈</mo> <mi>F</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mtext> </mtext> <mo>≤</mo> <mtext> </mtext> <mn>2</mn> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>m</mi> <mtext> </mtext> <mo><</mo> <mtext> </mtext> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{T\in F}\|T\|_{B(X,Y)}~\leq ~2\varepsilon ^{-1}m~<~\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eca644f1f8ca81a01a08a7674dcceac4aff9679" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.829ex; height:4.843ex;" alt="{\displaystyle \sup _{T\in F}\|T\|_{B(X,Y)}~\leq ~2\varepsilon ^{-1}m~<~\infty .}" /></span> </p> </div> <p>There are also simple proofs not using the Baire theorem (<a href="#CITEREFSokal2011">Sokal 2011</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Corollaries">Corollaries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_boundedness_principle&action=edit&section=2" title="Edit section: Corollaries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name">Corollary</strong><span class="theoreme-tiret"> — </span>If a sequence of bounded operators <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 \left(T_{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(T_{n}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6238e58fa37888ac6ffc61b61afa1cc8bc2481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.385ex; height:2.843ex;" alt="{\displaystyle \left(T_{n}\right)}" /></span> converges pointwise, that is, the limit of <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 \left(T_{n}(x)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(T_{n}(x)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f2f078020c0ce73c81a1326e33d385d844590de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.524ex; height:2.843ex;" alt="{\displaystyle \left(T_{n}(x)\right)}" /></span> exists for all <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\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" /></span> then these pointwise limits define a bounded linear operator <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de28b735beca1303bc5da9ba518a5a22a70a5d4" 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.}" /></span> </p> </div> <p>The above corollary does <em>not</em> claim that <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d9241493be76739f2400f258f32c24f9689161f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}" /></span> converges to <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" 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}" /></span> in operator norm, that is, uniformly on bounded sets. However, since <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 \left\{T_{n}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{T_{n}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4032c6f96f5b64b41d043fb36448ba151f239f07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.901ex; height:2.843ex;" alt="{\displaystyle \left\{T_{n}\right\}}" /></span> is bounded in operator norm, and the limit operator <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" 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}" /></span> is continuous, a standard "<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 3\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0903568bf6fb86b0008d9b6a2efc9a3a784642fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.246ex; height:2.176ex;" alt="{\displaystyle 3\varepsilon }" /></span>" estimate shows that <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d9241493be76739f2400f258f32c24f9689161f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}" /></span> converges to <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" 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}" /></span> uniformly on <em>compact</em> sets. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338" /><div class="math_proof" style=""><strong>Proof</strong> <p>Essentially the same as that of the proof that a pointwise convergent sequence of equicontinuous functions on a compact set converges to a continuous function. </p><p>By uniform boundedness principle, let <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 M=\max\{\sup _{n}\|T_{n}\|,\|T\|\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <mo fence="false" stretchy="false">‖</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\max\{\sup _{n}\|T_{n}\|,\|T\|\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b8d1de0175fb0868069f183683b15fefbcb9ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.976ex; height:4.343ex;" alt="{\displaystyle M=\max\{\sup _{n}\|T_{n}\|,\|T\|\}}" /></span> be a uniform upper bound on the operator norms. </p><p>Fix any compact <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 K\subset X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>⊂</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\subset X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feee4d451d1e9575b9be15f24b1c6cf37559037a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.144ex; height:2.176ex;" alt="{\displaystyle K\subset X}" /></span>. Then for any <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 \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}" /></span>, finitely cover (use compactness) <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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" /></span> by a finite set of open balls <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 \{B(x_{i},r)\}_{i=1,...,N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>B</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{B(x_{i},r)\}_{i=1,...,N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e49ec43a7b4dd08d6bda0d2982c012281a3248a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.756ex; height:3.009ex;" alt="{\displaystyle \{B(x_{i},r)\}_{i=1,...,N}}" /></span> of radius <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 r={\frac {\epsilon }{M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϵ</mi> <mi>M</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {\epsilon }{M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b681d5aeacf481612c1025c91bb0bca5c4a9c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.425ex; height:4.676ex;" alt="{\displaystyle r={\frac {\epsilon }{M}}}" /></span> </p><p>Since <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_{n}\to T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{n}\to T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cff7bbdca89ee847341519d1e584c4d70fa01fbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.826ex; height:2.509ex;" alt="{\displaystyle T_{n}\to T}" /></span> pointwise on each of <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_{1},...,x_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},...,x_{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c63e87a99ab8afe6960ef5908ce245629412d9a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.575ex; height:2.009ex;" alt="{\displaystyle x_{1},...,x_{N}}" /></span>, for all large <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>, <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_{n}(x_{i})-T(x_{i})\|\leq \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|T_{n}(x_{i})-T(x_{i})\|\leq \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3cbec81cfe900579dd4d6b91fb945ff60daacc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.298ex; height:2.843ex;" alt="{\displaystyle \|T_{n}(x_{i})-T(x_{i})\|\leq \epsilon }" /></span> for all <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 i=1,...,N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,...,N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3badf64e73cff33bd2c346727d32d72fa1f4e6ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.297ex; height:2.509ex;" alt="{\displaystyle i=1,...,N}" /></span>. </p><p>Then by triangle inequality, we find for all large <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>, <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 \forall x\in K,\|T_{n}(x)-T(x)\|\leq 3\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>K</mi> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mn>3</mn> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in K,\|T_{n}(x)-T(x)\|\leq 3\epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b03633a1f1413e37e97a8ed4d641f40d50fe4a23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.423ex; height:2.843ex;" alt="{\displaystyle \forall x\in K,\|T_{n}(x)-T(x)\|\leq 3\epsilon }" /></span>. </p> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name">Corollary</strong><span class="theoreme-tiret"> — </span>Any weakly bounded subset <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 S\subseteq Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0dcc51957b9926877744a0328f1a350d5ad07e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.371ex; height:2.343ex;" alt="{\displaystyle S\subseteq Y}" /></span> in a normed space <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> is bounded. </p> </div> <p>Indeed, the elements of <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> define a pointwise bounded family of continuous linear forms on the Banach space <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:=Y',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>:=</mo> <msup> <mi>Y</mi> <mo>′</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X:=Y',}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6103e446e8a1f0576f70232a28f94703edfb89ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.957ex; height:2.843ex;" alt="{\displaystyle X:=Y',}" /></span> which is the <a href="/wiki/Continuous_dual_space" class="mw-redirect" title="Continuous dual space">continuous dual space</a> of <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" /></span> By the uniform boundedness principle, the norms of elements of <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 S,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0bcd8516b165aaacb234616d7d2d23478a35be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.146ex; height:2.509ex;" alt="{\displaystyle S,}" /></span> as functionals on <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" /></span> that is, norms in the second dual <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'',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mo>″</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y'',}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c84f3eda6ca51727cd285671c4149cc0f3d1147" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.684ex; height:2.843ex;" alt="{\displaystyle Y'',}" /></span> are bounded. But for every <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 s\in S,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\in S,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad060d84dec285d2822233a418fbf98689f6ddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.077ex; height:2.509ex;" alt="{\displaystyle s\in S,}" /></span> the norm in the second dual coincides with the norm in <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3765557b7effa1a5f2f4dce9c80a25973b7009f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" /></span> by a consequence of the <a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach theorem</a>. </p><p>Let <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 L(X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d06d419d8c386f560a7de41b39972a6c55c55141" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.179ex; height:2.843ex;" alt="{\displaystyle L(X,Y)}" /></span> denote the continuous operators from <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> to <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3765557b7effa1a5f2f4dce9c80a25973b7009f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" /></span> endowed with the <a href="/wiki/Operator_norm" title="Operator norm">operator norm</a>. If the collection <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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" /></span> is unbounded in <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 L(X,Y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(X,Y),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4420840a22d8b5829f003c047417bbce5cf9c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.826ex; height:2.843ex;" alt="{\displaystyle L(X,Y),}" /></span> then the uniform boundedness principle implies: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\left\{x\in X\ :\ \sup \nolimits _{T\in F}\|Tx\|_{Y}=\infty \right\}\neq \varnothing .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <msub> <mo form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>∈</mo> <mi>F</mi> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo>}</mo> </mrow> <mo>≠</mo> <mi class="MJX-variant">∅</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\left\{x\in X\ :\ \sup \nolimits _{T\in F}\|Tx\|_{Y}=\infty \right\}\neq \varnothing .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82fed5720d5d1442e57a9bf8e55c588ab9f0ae57" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.181ex; height:2.843ex;" alt="{\displaystyle R=\left\{x\in X\ :\ \sup \nolimits _{T\in F}\|Tx\|_{Y}=\infty \right\}\neq \varnothing .}" /></span> </p><p>In fact, <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> is dense in <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" /></span> The complement of <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> is the countable union of closed sets <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="{\textstyle \bigcup X_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋃</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \bigcup X_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c820ef577e88e3632e4f63641de8399a0bfc0f06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.113ex; height:2.843ex;" alt="{\textstyle \bigcup X_{n}.}" /></span> By the argument used in proving the theorem, each <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}" /></span> is <a href="/wiki/Nowhere_dense" class="mw-redirect" title="Nowhere dense">nowhere dense</a>, i.e. the subset <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="{\textstyle \bigcup X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋃</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \bigcup X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af896b12b489dd5e6b73a7648fd03264a22055d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.466ex; height:2.843ex;" alt="{\textstyle \bigcup X_{n}}" /></span> is <em>of first category</em>. Therefore <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> is the complement of a subset of first category in a Baire space. By definition of a Baire space, such sets (called <em><a href="/wiki/Comeagre_set" class="mw-redirect" title="Comeagre set">comeagre</a></em> or <em>residual sets</em>) are dense. Such reasoning leads to the <em>principle of condensation of singularities</em>, which can be formulated as follows: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span>Let <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> be a Banach space, <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 \left(Y_{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(Y_{n}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58117b01f81e85adbed03c9cb0417e2636f6adbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle \left(Y_{n}\right)}" /></span> a sequence of normed vector spaces, and for every <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 n,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397bfafc701afdf14c2743278a097f6f2957eabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.042ex; height:2.009ex;" alt="{\displaystyle n,}" /></span> let <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 F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" /></span> an unbounded family in <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 L\left(X,Y_{n}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\left(X,Y_{n}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4d908249923c87816963459644122444d7f20e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.396ex; height:2.843ex;" alt="{\displaystyle L\left(X,Y_{n}\right).}" /></span> Then the set <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R:=\left\{x\in X\ :\ {\text{ for all }}n\in \mathbb {N} ,\sup _{T\in F_{n}}\|Tx\|_{Y_{n}}=\infty \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>:=</mo> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>∈</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R:=\left\{x\in X\ :\ {\text{ for all }}n\in \mathbb {N} ,\sup _{T\in F_{n}}\|Tx\|_{Y_{n}}=\infty \right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/418c0e14e45d4130cad202627e914051ee1bf853" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.703ex; height:6.343ex;" alt="{\displaystyle R:=\left\{x\in X\ :\ {\text{ for all }}n\in \mathbb {N} ,\sup _{T\in F_{n}}\|Tx\|_{Y_{n}}=\infty \right\}}" /></span> is a residual set, and thus dense in <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" /></span> </p> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338" /><div class="math_proof" style=""><strong>Proof</strong> <p>The complement of <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> is the countable union <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcup _{n,m}\left\{x\in X\ :\ \sup _{T\in F_{n}}\|Tx\|_{Y_{n}}\leq m\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </munder> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>∈</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>T</mi> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>≤</mo> <mi>m</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup _{n,m}\left\{x\in X\ :\ \sup _{T\in F_{n}}\|Tx\|_{Y_{n}}\leq m\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c10cbdf68f8436a257ff52621816fbf68780a829" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.64ex; height:6.843ex;" alt="{\displaystyle \bigcup _{n,m}\left\{x\in X\ :\ \sup _{T\in F_{n}}\|Tx\|_{Y_{n}}\leq m\right\}}" /></span> of sets of first category. Therefore, its residual set <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> is dense. </p> </div> <div class="mw-heading mw-heading2"><h2 id="Example:_pointwise_convergence_of_Fourier_series">Example: pointwise convergence of Fourier series</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_boundedness_principle&action=edit&section=3" title="Edit section: Example: pointwise convergence of Fourier series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <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 \mathbb {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c039979935c00b3b216cbb065999207872677f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} }" /></span> be the <a href="/wiki/Circle_group" title="Circle group">circle</a>, and let <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 C(\mathbb {T} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\mathbb {T} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64fe8e0057b2c3866712eba11b39e6d4583a9fc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.126ex; height:2.843ex;" alt="{\displaystyle C(\mathbb {T} )}" /></span> be the Banach space of continuous functions on <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 \mathbb {T} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b582671af3fc6119dbab528faeeb412844442931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.197ex; height:2.509ex;" alt="{\displaystyle \mathbb {T} ,}" /></span> with the <a href="/wiki/Uniform_norm" title="Uniform norm">uniform norm</a>. Using the uniform boundedness principle, one can show that there exists an element in <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 C(\mathbb {T} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\mathbb {T} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64fe8e0057b2c3866712eba11b39e6d4583a9fc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.126ex; height:2.843ex;" alt="{\displaystyle C(\mathbb {T} )}" /></span> for which the Fourier series does not converge pointwise. </p><p>For <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 f\in C(\mathbb {T} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in C(\mathbb {T} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe735c39daff2b40895f3342ab528e91d704e39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.892ex; height:2.843ex;" alt="{\displaystyle f\in C(\mathbb {T} ),}" /></span> its <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> is defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k\in \mathbb {Z} }{\hat {f}}(k)e^{ikx}=\sum _{k\in \mathbb {Z} }{\frac {1}{2\pi }}\left(\int _{0}^{2\pi }f(t)e^{-ikt}dt\right)e^{ikx},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mi>t</mi> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k\in \mathbb {Z} }{\hat {f}}(k)e^{ikx}=\sum _{k\in \mathbb {Z} }{\frac {1}{2\pi }}\left(\int _{0}^{2\pi }f(t)e^{-ikt}dt\right)e^{ikx},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b2c1b0d64046109428b35f9c6baeb7f8297f17f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.277ex; height:7.009ex;" alt="{\displaystyle \sum _{k\in \mathbb {Z} }{\hat {f}}(k)e^{ikx}=\sum _{k\in \mathbb {Z} }{\frac {1}{2\pi }}\left(\int _{0}^{2\pi }f(t)e^{-ikt}dt\right)e^{ikx},}" /></span> and the <i>N</i>-th symmetric partial sum is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{N}(f)(x)=\sum _{k=-N}^{N}{\hat {f}}(k)e^{ikx}={\frac {1}{2\pi }}\int _{0}^{2\pi }f(t)D_{N}(x-t)\,dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{N}(f)(x)=\sum _{k=-N}^{N}{\hat {f}}(k)e^{ikx}={\frac {1}{2\pi }}\int _{0}^{2\pi }f(t)D_{N}(x-t)\,dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6422f2ecda7b2a9af0ee02b01a1adccd70dd175" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.216ex; height:7.509ex;" alt="{\displaystyle S_{N}(f)(x)=\sum _{k=-N}^{N}{\hat {f}}(k)e^{ikx}={\frac {1}{2\pi }}\int _{0}^{2\pi }f(t)D_{N}(x-t)\,dt,}" /></span> where <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_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69fcf2425a1e9f9ef9979b837f32bdfbcd4beff5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.616ex; height:2.509ex;" alt="{\displaystyle D_{N}}" /></span> is the <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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /></span>-th <a href="/wiki/Dirichlet_kernel" title="Dirichlet kernel">Dirichlet kernel</a>. Fix <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\in \mathbb {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d049929c2e82026ec84d3c2300c3fc905dce4633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.721ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {T} }" /></span> and consider the convergence of <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 \left\{S_{N}(f)(x)\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{S_{N}(f)(x)\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc0788b7652957a7a47255951503c4dda8c169b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.702ex; height:2.843ex;" alt="{\displaystyle \left\{S_{N}(f)(x)\right\}.}" /></span> The functional <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 \varphi _{N,x}:C(\mathbb {T} )\to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>x</mi> </mrow> </msub> <mo>:</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{N,x}:C(\mathbb {T} )\to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce2b145a725c9141bcd3755dba5dd470a739bf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.964ex; height:3.009ex;" alt="{\displaystyle \varphi _{N,x}:C(\mathbb {T} )\to \mathbb {C} }" /></span> defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{N,x}(f)=S_{N}(f)(x),\qquad f\in C(\mathbb {T} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em"></mspace> <mi>f</mi> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{N,x}(f)=S_{N}(f)(x),\qquad f\in C(\mathbb {T} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64008973d0a74b6d0d454a7bf558dffcccddfd13" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.71ex; height:3.009ex;" alt="{\displaystyle \varphi _{N,x}(f)=S_{N}(f)(x),\qquad f\in C(\mathbb {T} ),}" /></span> is bounded. The norm of <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 \varphi _{N,x},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>x</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{N,x},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33deafcbe90449cf0a53aee9370ece9036e07551" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.256ex; height:2.343ex;" alt="{\displaystyle \varphi _{N,x},}" /></span> in the dual of <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 C(\mathbb {T} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\mathbb {T} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/254d0a6a57f1f33c8cac4d20b5d2ddeabd8294f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.773ex; height:2.843ex;" alt="{\displaystyle C(\mathbb {T} ),}" /></span> is the norm of the signed measure <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 (2(2\pi )^{-1}D_{N}(x-t)dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2(2\pi )^{-1}D_{N}(x-t)dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8c7390c6429a78c7568d3ec1b0bfc8d0cc0899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.841ex; height:3.176ex;" alt="{\displaystyle (2(2\pi )^{-1}D_{N}(x-t)dt,}" /></span> namely <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\|\varphi _{N,x}\right\|={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|D_{N}(x-t)\right|\,dt={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|D_{N}(s)\right|\,ds=\left\|D_{N}\right\|_{L^{1}(\mathbb {T} )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>x</mi> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mrow> <mo>|</mo> <mrow> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mrow> <mo>|</mo> <mrow> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>s</mi> <mo>=</mo> <msub> <mrow> <mo symmetric="true">‖</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|\varphi _{N,x}\right\|={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|D_{N}(x-t)\right|\,dt={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|D_{N}(s)\right|\,ds=\left\|D_{N}\right\|_{L^{1}(\mathbb {T} )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3c162e6583ded595011d7026c05c14a1dfe315" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:69.954ex; height:6.176ex;" alt="{\displaystyle \left\|\varphi _{N,x}\right\|={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|D_{N}(x-t)\right|\,dt={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|D_{N}(s)\right|\,ds=\left\|D_{N}\right\|_{L^{1}(\mathbb {T} )}.}" /></span> </p><p>It can be verified that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2\pi }}\int _{0}^{2\pi }|D_{N}(t)|\,dt\geq {\frac {1}{2\pi }}\int _{0}^{2\pi }{\frac {\left|\sin \left((N+{\tfrac {1}{2}})t\right)\right|}{t/2}}\,dt\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi }}\int _{0}^{2\pi }|D_{N}(t)|\,dt\geq {\frac {1}{2\pi }}\int _{0}^{2\pi }{\frac {\left|\sin \left((N+{\tfrac {1}{2}})t\right)\right|}{t/2}}\,dt\to \infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e008634f407325c420c1b9838e3e770a980603a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:55.583ex; height:7.176ex;" alt="{\displaystyle {\frac {1}{2\pi }}\int _{0}^{2\pi }|D_{N}(t)|\,dt\geq {\frac {1}{2\pi }}\int _{0}^{2\pi }{\frac {\left|\sin \left((N+{\tfrac {1}{2}})t\right)\right|}{t/2}}\,dt\to \infty .}" /></span> </p><p>So the collection <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 \left(\varphi _{N,x}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\varphi _{N,x}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1138fa0bd3cf533762086cd9d29793a6057590a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.419ex; height:3.009ex;" alt="{\displaystyle \left(\varphi _{N,x}\right)}" /></span> is unbounded in <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 C(\mathbb {T} )^{\ast },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\mathbb {T} )^{\ast },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/643f15b3a963850d055c34b0a1eae512f52a871b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.827ex; height:2.843ex;" alt="{\displaystyle C(\mathbb {T} )^{\ast },}" /></span> the dual of <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 C(\mathbb {T} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\mathbb {T} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e683e9f1f7eeb3e5b4da56131166897b250d3794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.773ex; height:2.843ex;" alt="{\displaystyle C(\mathbb {T} ).}" /></span> Therefore, by the uniform boundedness principle, for any <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\in \mathbb {T} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {T} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4177694ea0202d49708817a87041425a28382912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.368ex; height:2.509ex;" alt="{\displaystyle x\in \mathbb {T} ,}" /></span> the set of continuous functions whose Fourier series diverges at <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}" /></span> is dense in <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 C(\mathbb {T} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\mathbb {T} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e683e9f1f7eeb3e5b4da56131166897b250d3794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.773ex; height:2.843ex;" alt="{\displaystyle C(\mathbb {T} ).}" /></span> </p><p>More can be concluded by applying the principle of condensation of singularities. Let <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 \left(x_{m}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x_{m}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6932305c9878cd44dfbc903b68227fb3e04fb87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.814ex; height:2.843ex;" alt="{\displaystyle \left(x_{m}\right)}" /></span> be a dense sequence in <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 \mathbb {T} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23bfcd3d5e94ad3ec191037d3e330ea77943d9cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} .}" /></span> Define <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 \varphi _{N,x_{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{N,x_{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71cf5fe5499f0a2b35b165b99db2bce1f53f7a78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.945ex; height:2.343ex;" alt="{\displaystyle \varphi _{N,x_{m}}}" /></span> in the similar way as above. The principle of condensation of singularities then says that the set of continuous functions whose Fourier series diverges at each <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_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f12100e1dc5769ced8c9806b219abc06ab321d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.005ex; height:2.009ex;" alt="{\displaystyle x_{m}}" /></span> is dense in <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 C(\mathbb {T} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\mathbb {T} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64fe8e0057b2c3866712eba11b39e6d4583a9fc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.126ex; height:2.843ex;" alt="{\displaystyle C(\mathbb {T} )}" /></span> (however, the Fourier series of a continuous function <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}" /></span> converges to <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 f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" 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)}" /></span> for almost every <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\in \mathbb {T} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {T} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4177694ea0202d49708817a87041425a28382912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.368ex; height:2.509ex;" alt="{\displaystyle x\in \mathbb {T} ,}" /></span> by <a href="/wiki/Carleson%27s_theorem" title="Carleson's theorem">Carleson's theorem</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_boundedness_principle&action=edit&section=4" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</a> (TVS) <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" /></span> "bounded subset" refers specifically to the notion of a <a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">von Neumann bounded subset</a>. If <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> happens to also be a normed or <a href="/wiki/Seminormed_space" class="mw-redirect" title="Seminormed space">seminormed space</a>, say with <a href="/wiki/Seminorm" title="Seminorm">(semi)norm</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 \|\cdot \|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de96e8c0ebaff5cc1b2a201f77e17701323191e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.651ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|,}" /></span> then a subset <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span> is (von Neumann) bounded if and only if it is <em>norm bounded</em>, which by definition means <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="{\textstyle \sup _{b\in B}\|b\|<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>∈</mo> <mi>B</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>b</mi> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sup _{b\in B}\|b\|<\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d84ff0e2877a72c07e15d30bc3bfaaf6a441bc5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.561ex; height:2.843ex;" alt="{\textstyle \sup _{b\in B}\|b\|<\infty .}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Barrelled_spaces">Barrelled spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_boundedness_principle&action=edit&section=5" title="Edit section: Barrelled spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled space</a></div> <p>Attempts to find classes of <a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">locally convex topological vector spaces</a> on which the uniform boundedness principle holds eventually led to <a href="/wiki/Barrelled_space" title="Barrelled space">barrelled spaces</a>. That is, the least restrictive setting for the uniform boundedness principle is a barrelled space, where the following generalized version of the theorem holds (<a href="#CITEREFBourbaki1987">Bourbaki 1987</a>, Theorem III.2.1): </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span>Given a barrelled space <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> and a <a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">locally convex space</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 Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3765557b7effa1a5f2f4dce9c80a25973b7009f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" /></span> then any family of pointwise bounded <a href="/wiki/Continuous_linear_mapping" class="mw-redirect" title="Continuous linear mapping">continuous linear mappings</a> from <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> to <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> is <a href="/wiki/Equicontinuous" class="mw-redirect" title="Equicontinuous">equicontinuous</a> (and even <a href="/wiki/Uniformly_equicontinuous" class="mw-redirect" title="Uniformly equicontinuous">uniformly equicontinuous</a>). </p><p>Alternatively, the statement also holds whenever <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> is a <a href="/wiki/Baire_space" title="Baire space">Baire space</a> and <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> is a locally convex space.<sup id="cite_ref-FOOTNOTEShtern2001_1-0" class="reference"><a href="#cite_note-FOOTNOTEShtern2001-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> </div> <div class="mw-heading mw-heading3"><h3 id="Uniform_boundedness_in_topological_vector_spaces">Uniform boundedness in topological vector spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_boundedness_principle&action=edit&section=6" title="Edit section: Uniform boundedness in topological vector spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Uniformly_bounded_sets_(topological_vector_space)" class="mw-redirect" title="Uniformly bounded sets (topological vector space)">Uniformly bounded sets (topological vector space)</a></div> <p>A <a href="/wiki/Family_of_sets" title="Family of sets">family</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 {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5622de88a69f68340f8dcb43d0b8bd443ba9e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" /></span> of subsets of a <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> is said to be <em><a href="/wiki/Uniformly_bounded_sets_(topological_vector_space)" class="mw-redirect" title="Uniformly bounded sets (topological vector space)">uniformly bounded</a></em> in <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3765557b7effa1a5f2f4dce9c80a25973b7009f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" /></span> if there exists some <a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">bounded subset</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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" /></span> of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq D\quad {\text{ for every }}B\in {\mathcal {B}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊆</mo> <mi>D</mi> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> for every </mtext> </mrow> <mi>B</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subseteq D\quad {\text{ for every }}B\in {\mathcal {B}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bee21ec6c0c8da5bae2ec6ba6a8413e5c5af3f3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.863ex; height:2.509ex;" alt="{\displaystyle B\subseteq D\quad {\text{ for every }}B\in {\mathcal {B}},}" /></span> which happens if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcup _{B\in {\mathcal {B}}}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mrow> </munder> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup _{B\in {\mathcal {B}}}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9455ea3c6fa3783128e9a0bb751df65d83e8d3a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:5.586ex; height:5.676ex;" alt="{\displaystyle \bigcup _{B\in {\mathcal {B}}}B}" /></span> is a bounded subset of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span>; if <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> is a <a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">normed space</a> then this happens if and only if there exists some real <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 M\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/069c80bca00b08466c7ef8ce37a46a56035dcef8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.703ex; height:2.343ex;" alt="{\displaystyle M\geq 0}" /></span> such that <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="{\textstyle \sup _{\stackrel {b\in B}{B\in {\mathcal {B}}}}\|b\|\leq M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>B</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>∈</mo> <mi>B</mi> </mrow> </mover> </mrow> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>b</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sup _{\stackrel {b\in B}{B\in {\mathcal {B}}}}\|b\|\leq M.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1279c9f873b7164619ab42a4bdf95142b6c5c23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.066ex; height:4.176ex;" alt="{\textstyle \sup _{\stackrel {b\in B}{B\in {\mathcal {B}}}}\|b\|\leq M.}" /></span> In particular, if <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /></span> is a family of maps from <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> to <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> and if <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 C\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a98beab5a3637ffa2321ef617c02a3b5b26ffd3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.845ex; height:2.343ex;" alt="{\displaystyle C\subseteq X}" /></span> then the family <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 \{h(C):h\in H\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>h</mi> <mo>∈</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{h(C):h\in H\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d78886915ad4a3b36d630a7b3524453463201a0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.42ex; height:2.843ex;" alt="{\displaystyle \{h(C):h\in H\}}" /></span> is uniformly bounded in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> if and only if there exists some bounded subset <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" /></span> of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> such that <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 h(C)\subseteq D{\text{ for all }}h\in H,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>h</mi> <mo>∈</mo> <mi>H</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(C)\subseteq D{\text{ for all }}h\in H,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65f6627c29f6530abacdbbc92259a634763120ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.811ex; height:2.843ex;" alt="{\displaystyle h(C)\subseteq D{\text{ for all }}h\in H,}" /></span> which happens if and only if <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="{\textstyle H(C):=\bigcup _{h\in H}h(C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>∈</mo> <mi>H</mi> </mrow> </munder> <mi>h</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle H(C):=\bigcup _{h\in H}h(C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/723108ba56f8026a3fe2241f8b035467f4fcf6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.357ex; height:3.009ex;" alt="{\textstyle H(C):=\bigcup _{h\in H}h(C)}" /></span> is a bounded subset of <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" /></span> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name">Proposition<sup id="cite_ref-FOOTNOTERudin199142−47_2-0" class="reference"><a href="#cite_note-FOOTNOTERudin199142−47-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></strong><span class="theoreme-tiret"> — </span>Let <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 H\subseteq L(X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>⊆</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\subseteq L(X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48336f24bfeb89b5868268e742d1d7ded8b37b24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.341ex; height:2.843ex;" alt="{\displaystyle H\subseteq L(X,Y)}" /></span> be a set of continuous linear operators between two <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector spaces</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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> and <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> and let <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 C\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a98beab5a3637ffa2321ef617c02a3b5b26ffd3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.845ex; height:2.343ex;" alt="{\displaystyle C\subseteq X}" /></span> be any <a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">bounded subset</a> of <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" /></span> Then the <a href="/wiki/Family_of_sets" title="Family of sets">family of sets</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 \{h(C):h\in H\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>h</mi> <mo>∈</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{h(C):h\in H\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d78886915ad4a3b36d630a7b3524453463201a0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.42ex; height:2.843ex;" alt="{\displaystyle \{h(C):h\in H\}}" /></span> is uniformly bounded in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> if any of the following conditions are satisfied: </p> <ol><li><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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /></span> is equicontinuous.</li> <li><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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> is a <a href="/wiki/Convex_set" title="Convex set">convex</a> <a href="/wiki/Compact_space" title="Compact space">compact</a> Hausdorff <a href="/wiki/Subspace_(topology)" class="mw-redirect" title="Subspace (topology)">subspace</a> of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> and for every <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 c\in C,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mi>C</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in C,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e13ac2b1dda38dca38dd73c97f0ee57023f3be9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.261ex; height:2.509ex;" alt="{\displaystyle c\in C,}" /></span> the orbit <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 H(c):=\{h(c):h\in H\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>h</mi> <mo>∈</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(c):=\{h(c):h\in H\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59beea6dc9eac775de5fda1cd291397aa7f74d46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.286ex; height:2.843ex;" alt="{\displaystyle H(c):=\{h(c):h\in H\}}" /></span> is a bounded subset of <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" /></span></li></ol> </div> <div class="mw-heading mw-heading3"><h3 id="Generalizations_involving_nonmeager_subsets">Generalizations involving nonmeager subsets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_boundedness_principle&action=edit&section=7" title="Edit section: Generalizations involving nonmeager subsets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although the notion of a <a href="/wiki/Nonmeager_set" class="mw-redirect" title="Nonmeager set">nonmeager set</a> is used in the following version of the uniform bounded principle, the domain <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> is <em>not</em> assumed to be a <a href="/wiki/Baire_space" title="Baire space">Baire space</a>. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem<sup id="cite_ref-FOOTNOTERudin199142−47_2-1" class="reference"><a href="#cite_note-FOOTNOTERudin199142−47-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></strong><span class="theoreme-tiret"> — </span>Let <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 H\subseteq L(X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>⊆</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\subseteq L(X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48336f24bfeb89b5868268e742d1d7ded8b37b24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.341ex; height:2.843ex;" alt="{\displaystyle H\subseteq L(X,Y)}" /></span> be a set of <a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">continuous linear operators</a> between two <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector spaces</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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> and <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> (not necessarily <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> or locally convex). For every <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\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" /></span> denote the <a href="/wiki/Orbit_(group_theory)" class="mw-redirect" title="Orbit (group theory)">orbit</a> of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}" /></span> by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x):=\{h(x):h\in H\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>h</mi> <mo>∈</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x):=\{h(x):h\in H\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5a48fe43e6006f982590bd2c3433846c2ef379" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.931ex; height:2.843ex;" alt="{\displaystyle H(x):=\{h(x):h\in H\}}" /></span> and let <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span> denote the set of all <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\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" /></span> whose orbit <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 H(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11f241aa7195bebab9d0a3c248ea97ef0c78b1ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.203ex; height:2.843ex;" alt="{\displaystyle H(x)}" /></span> is a <a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">bounded subset</a> of <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" /></span> If <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span> is of the <a href="/wiki/Second_category" class="mw-redirect" title="Second category">second category</a> (that is, nonmeager) in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> then <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 B=X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/764e9d661309f4d92ca2e43d1aa6573f92a12dd4" 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 B=X}" /></span> and <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /></span> is equicontinuous. </p> </div> <p>Every proper vector subspace of a TVS <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> has an empty interior in <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" /></span><sup id="cite_ref-FOOTNOTERudin199146_3-0" class="reference"><a href="#cite_note-FOOTNOTERudin199146-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> So in particular, every proper vector subspace that is closed is nowhere dense in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> and thus of the first category (meager) in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> (and the same is thus also true of all its subsets). Consequently, any vector subspace of a TVS <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> that is of the second category (nonmeager) in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> must be a <a href="/wiki/Dense_set" title="Dense set">dense subset</a> of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> (since otherwise its closure in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> would a closed proper vector subspace of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> and thus of the first category).<sup id="cite_ref-FOOTNOTERudin199146_3-1" class="reference"><a href="#cite_note-FOOTNOTERudin199146-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338" /><div class="math_proof" style=""><strong>Proof<sup id="cite_ref-FOOTNOTERudin199142−47_2-2" class="reference"><a href="#cite_note-FOOTNOTERudin199142−47-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></strong> <p><em>Proof that <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /></span> is equicontinuous:</em> </p><p>Let <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 W,V\subseteq Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>,</mo> <mi>V</mi> <mo>⊆</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W,V\subseteq Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5040e2de5a201e758558e08b032a9971eedaac1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.128ex; height:2.509ex;" alt="{\displaystyle W,V\subseteq Y}" /></span> be <a href="/wiki/Balanced_set" title="Balanced set">balanced</a> neighborhoods of the origin in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> satisfying <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 {\overline {V}}+{\overline {V}}\subseteq W.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>⊆</mo> <mi>W</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {V}}+{\overline {V}}\subseteq W.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/133d01e8349ca02ed7ca6dd10fafc365fcaf95f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.387ex; height:3.176ex;" alt="{\displaystyle {\overline {V}}+{\overline {V}}\subseteq W.}" /></span> It must be shown that there exists a neighborhood <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 N\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87353e8ceae2fe8f047c233b269f03d0cb80da93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.142ex; height:2.343ex;" alt="{\displaystyle N\subseteq X}" /></span> of the origin in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> such that <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 h(N)\subseteq W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(N)\subseteq W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1fe23221519ec69176c2aeecaeec6507df0eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.746ex; height:2.843ex;" alt="{\displaystyle h(N)\subseteq W}" /></span> for every <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 h\in H.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∈</mo> <mi>H</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\in H.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d10c356f8b2cea62359891c45c61a5670d8ade85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.89ex; height:2.176ex;" alt="{\displaystyle h\in H.}" /></span> Let <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C~:=~\bigcap _{h\in H}h^{-1}\left({\overline {V}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mtext> </mtext> <mo>:=</mo> <mtext> </mtext> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>∈</mo> <mi>H</mi> </mrow> </munder> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C~:=~\bigcap _{h\in H}h^{-1}\left({\overline {V}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc6edd3e14d2b320dbc6be25f20e6b792b31895" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.001ex; height:6.176ex;" alt="{\displaystyle C~:=~\bigcap _{h\in H}h^{-1}\left({\overline {V}}\right),}" /></span> which is a closed subset of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> (because it is an intersection of closed subsets) that for every <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 h\in H,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∈</mo> <mi>H</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\in H,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ad91aec7b3a20bc5c6ca74a374ef9032b1d8be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.89ex; height:2.509ex;" alt="{\displaystyle h\in H,}" /></span> also satisfies <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 h(C)\subseteq {\overline {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(C)\subseteq {\overline {V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d84baf674bd46b7282d4d8ac20905f0e100433" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.196ex; height:3.509ex;" alt="{\displaystyle h(C)\subseteq {\overline {V}}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(C-C)~=~h(C)-h(C)~\subseteq ~{\overline {V}}-{\overline {V}}~=~{\overline {V}}+{\overline {V}}~\subseteq ~W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo>−</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>h</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo accent="false">¯</mo> </mover> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo accent="false">¯</mo> </mover> </mrow> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(C-C)~=~h(C)-h(C)~\subseteq ~{\overline {V}}-{\overline {V}}~=~{\overline {V}}+{\overline {V}}~\subseteq ~W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d81f9abccff399925c0bcd55621d54bb1a70410" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.078ex; height:3.509ex;" alt="{\displaystyle h(C-C)~=~h(C)-h(C)~\subseteq ~{\overline {V}}-{\overline {V}}~=~{\overline {V}}+{\overline {V}}~\subseteq ~W}" /></span> (as will be shown, the set <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 C-C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>−</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C-C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a81c013f8303e973aab22cb215624529f3b4c26d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.373ex; height:2.343ex;" alt="{\displaystyle C-C}" /></span> is in fact a neighborhood of the origin in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> because the topological interior of <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> is not empty). If <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 b\in B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∈</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\in B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61dbfba9ff608c8700a30596649d98dcc6147d86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle b\in B}" /></span> then <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 H(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/810ab14fd6bf72cf37e744fe2f1dce945b78d7b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.87ex; height:2.843ex;" alt="{\displaystyle H(b)}" /></span> being bounded in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> implies that there exists some integer <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 n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" /></span> such that <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 H(b)\subseteq nV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi>n</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(b)\subseteq nV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/743882e61d55ed56344831a2162982da767c5939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.151ex; height:2.843ex;" alt="{\displaystyle H(b)\subseteq nV}" /></span> so if <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 h\in H,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∈</mo> <mi>H</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\in H,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ad91aec7b3a20bc5c6ca74a374ef9032b1d8be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.89ex; height:2.509ex;" alt="{\displaystyle h\in H,}" /></span> then <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 b~\in ~h^{-1}\left(nV\right)~=~nh^{-1}(V).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mtext> </mtext> <mo>∈</mo> <mtext> </mtext> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mi>V</mi> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>n</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b~\in ~h^{-1}\left(nV\right)~=~nh^{-1}(V).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae29ca75ed0c1bd8972069e38e6561c9ac9b556e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.006ex; height:3.176ex;" alt="{\displaystyle b~\in ~h^{-1}\left(nV\right)~=~nh^{-1}(V).}" /></span> Since <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 h\in H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∈</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\in H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/675a79e26028d91d97f4e2ce279c314b0f194c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.243ex; height:2.176ex;" alt="{\displaystyle h\in H}" /></span> was arbitrary, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b~\in ~\bigcap _{h\in H}nh^{-1}(V)~=~n\bigcap _{h\in H}h^{-1}(V)~\subseteq ~nC.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mtext> </mtext> <mo>∈</mo> <mtext> </mtext> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>∈</mo> <mi>H</mi> </mrow> </munder> <mi>n</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>n</mi> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>∈</mo> <mi>H</mi> </mrow> </munder> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <mi>n</mi> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b~\in ~\bigcap _{h\in H}nh^{-1}(V)~=~n\bigcap _{h\in H}h^{-1}(V)~\subseteq ~nC.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9abed8c696a34e9223f51b8bb3d3a05bd23af6d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.206ex; height:5.676ex;" alt="{\displaystyle b~\in ~\bigcap _{h\in H}nh^{-1}(V)~=~n\bigcap _{h\in H}h^{-1}(V)~\subseteq ~nC.}" /></span> This proves that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B~\subseteq ~\bigcup _{n\in \mathbb {N} }nC.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <mi>n</mi> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B~\subseteq ~\bigcup _{n\in \mathbb {N} }nC.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/126477adaf4cfecd69f494ef0503453304ed28a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.875ex; height:5.676ex;" alt="{\displaystyle B~\subseteq ~\bigcup _{n\in \mathbb {N} }nC.}" /></span> Because <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span> is of the second category in <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" /></span> the same must be true of at least one of the sets <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 nC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle nC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b586b1549993466229bb5bb0d266a04e0c0f8bb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.161ex; height:2.176ex;" alt="{\displaystyle nC}" /></span> for some <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 n\in \mathbb {N} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b82f4024a32a795b10f00971e03ad88ebaa367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.56ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} .}" /></span> The map <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\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83d8a6029587ee9b365bdeab1e2f4b7c469b0219" 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}" /></span> defined by <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="{\textstyle x\mapsto {\frac {1}{n}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x\mapsto {\frac {1}{n}}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05b7ba3a6e3dc377d2fc3d2f1f1566c4e56422ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.096ex; height:3.343ex;" alt="{\textstyle x\mapsto {\frac {1}{n}}x}" /></span> is a (<a href="/wiki/Surjective_function" title="Surjective function">surjective</a>) <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a>, so the set <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="{\textstyle {\frac {1}{n}}(nC)=C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{n}}(nC)=C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9c6aba5aeaef7bc70a745fb3519391a7149641" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.657ex; height:3.343ex;" alt="{\textstyle {\frac {1}{n}}(nC)=C}" /></span> is necessarily of the second category in <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" /></span> Because <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> is closed and of the second category in <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" /></span> its <a href="/wiki/Topological_interior" class="mw-redirect" title="Topological interior">topological interior</a> in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> is not empty. Pick <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 c\in \operatorname {Int} _{X}C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in \operatorname {Int} _{X}C.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c4991d1b9cb6bb978dee75316807d8d29832cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.317ex; height:2.509ex;" alt="{\displaystyle c\in \operatorname {Int} _{X}C.}" /></span> Because the map <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\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83d8a6029587ee9b365bdeab1e2f4b7c469b0219" 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}" /></span> defined by <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\mapsto c-x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>c</mi> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto c-x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eae78c484db504aa86eb8bc887331205bc2fa385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.121ex; height:2.176ex;" alt="{\displaystyle x\mapsto c-x}" /></span> is a homeomorphism, the set <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N~:=~c-\operatorname {Int} _{X}C~=~\operatorname {Int} _{X}(c-C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mtext> </mtext> <mo>:=</mo> <mtext> </mtext> <mi>c</mi> <mo>−</mo> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>C</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N~:=~c-\operatorname {Int} _{X}C~=~\operatorname {Int} _{X}(c-C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d76f09378b80607de5933debbe0bfb9e9b8d76a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.379ex; height:2.843ex;" alt="{\displaystyle N~:=~c-\operatorname {Int} _{X}C~=~\operatorname {Int} _{X}(c-C)}" /></span> is a neighborhood of <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 0=c-c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mi>c</mi> <mo>−</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=c-c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1beff98c15a5089580eb4fb402960ac643ef7e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.115ex; height:2.343ex;" alt="{\displaystyle 0=c-c}" /></span> in <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" /></span> which implies that the same is true of its superset <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 C-C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>−</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C-C.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79bb73bba5cccf0216f292d60208f418deb06318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.02ex; height:2.343ex;" alt="{\displaystyle C-C.}" /></span> And so for every <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 h\in H,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∈</mo> <mi>H</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\in H,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ad91aec7b3a20bc5c6ca74a374ef9032b1d8be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.89ex; height:2.509ex;" alt="{\displaystyle h\in H,}" /></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(N)~\subseteq ~h(c-C)~=~h(c)-h(C)~\subseteq ~{\overline {V}}-{\overline {V}}~\subseteq ~W.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <mi>h</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>h</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo accent="false">¯</mo> </mover> </mrow> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <mi>W</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(N)~\subseteq ~h(c-C)~=~h(c)-h(C)~\subseteq ~{\overline {V}}-{\overline {V}}~\subseteq ~W.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62d65713ffde211e1d50d8030e44de4480fde329" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.211ex; height:3.509ex;" alt="{\displaystyle h(N)~\subseteq ~h(c-C)~=~h(c)-h(C)~\subseteq ~{\overline {V}}-{\overline {V}}~\subseteq ~W.}" /></span> This proves that <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /></span> is equicontinuous. Q.E.D. </p> <hr style="height:1px" /> <p><em>Proof that <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 B=X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/764e9d661309f4d92ca2e43d1aa6573f92a12dd4" 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 B=X}" /></span>:</em> </p><p>Because <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /></span> is equicontinuous, if <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 S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}" /></span> is bounded in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> then <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 H(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1061a0762c27da11a9f4a68b4775d3273f6c9d2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.372ex; height:2.843ex;" alt="{\displaystyle H(S)}" /></span> is uniformly bounded in <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" /></span> In particular, for any <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\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" /></span> because <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 S:=\{x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S:=\{x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce91715f9177f18987d6d0cd06fc84dfe36b39a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.899ex; height:2.843ex;" alt="{\displaystyle S:=\{x\}}" /></span> is a bounded subset of <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" /></span> <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 H(\{x\})=H(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(\{x\})=H(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f9f371ceb7229d975b41ce92a92a6081558c4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.829ex; height:2.843ex;" alt="{\displaystyle H(\{x\})=H(x)}" /></span> is a uniformly bounded subset of <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" /></span> Thus <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 B=X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b137bb362c523aa15e77250192d7b261b48f2b3e" 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 B=X.}" /></span> Q.E.D. </p> </div> <div class="mw-heading mw-heading3"><h3 id="Sequences_of_continuous_linear_maps">Sequences of continuous linear maps</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_boundedness_principle&action=edit&section=8" title="Edit section: Sequences of continuous linear maps"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following theorem establishes conditions for the pointwise limit of a sequence of continuous linear maps to be itself continuous. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem<sup id="cite_ref-FOOTNOTERudin199145−46_4-0" class="reference"><a href="#cite_note-FOOTNOTERudin199145−46-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></strong><span class="theoreme-tiret"> — </span>Suppose that <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 h_{1},h_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{1},h_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8934cba37bf9026a3b42d3d7cca5f05d3c726b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.578ex; height:2.509ex;" alt="{\displaystyle h_{1},h_{2},\ldots }" /></span> is a sequence of continuous linear maps between two <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector spaces</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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> and <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" /></span> </p> <ol><li>If the set <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> of all <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\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" /></span> for which <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 h_{1}(x),h_{2}(x),\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{1}(x),h_{2}(x),\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b09b0d7a7ff2f52d3e083b0217e5c99a0a6ee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.855ex; height:2.843ex;" alt="{\displaystyle h_{1}(x),h_{2}(x),\ldots }" /></span> is a Cauchy sequence in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> is of the second category in <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" /></span> then <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 C=X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22eacb4374a77c1272969a123b43e2adce7c61e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.492ex; height:2.176ex;" alt="{\displaystyle C=X.}" /></span></li> <li>If the set <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" /></span> of all <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\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" /></span> at which the limit <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 h(x):=\lim _{n\to \infty }h_{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x):=\lim _{n\to \infty }h_{n}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab165109011b60969431e9c3c7337b22de985145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.579ex; height:3.843ex;" alt="{\displaystyle h(x):=\lim _{n\to \infty }h_{n}(x)}" /></span> exists in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> is of the second category in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> and if <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> is a <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">complete</a> <a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">metrizable topological vector space</a> (such as a <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet space</a> or an <a href="/wiki/F-space" title="F-space">F-space</a>), then <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 L=X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77e1a77fccaa0d67f0440b0fa9bb060f8fbf1b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.661ex; height:2.176ex;" alt="{\displaystyle L=X}" /></span> and <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 h:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d91a802daa0df93b0c6fcc9e2065fac31ce08e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.644ex; height:2.176ex;" alt="{\displaystyle h:X\to Y}" /></span> is a continuous linear map.</li></ol> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem<sup id="cite_ref-FOOTNOTERudin199146_3-2" class="reference"><a href="#cite_note-FOOTNOTERudin199146-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></strong><span class="theoreme-tiret"> — </span>If <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 h_{1},h_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{1},h_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8934cba37bf9026a3b42d3d7cca5f05d3c726b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.578ex; height:2.509ex;" alt="{\displaystyle h_{1},h_{2},\ldots }" /></span> is a sequence of continuous linear maps from an <a href="/wiki/F-space" title="F-space">F-space</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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> into a Hausdorff topological vector space <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> such that for every <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\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" /></span> the limit <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)~:=~\lim _{n\to \infty }h_{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>:=</mo> <mtext> </mtext> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)~:=~\lim _{n\to \infty }h_{n}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aacc6abe2e5149808f2389ca1e4bf615542969" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.128ex; height:3.843ex;" alt="{\displaystyle h(x)~:=~\lim _{n\to \infty }h_{n}(x)}" /></span> exists in <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3765557b7effa1a5f2f4dce9c80a25973b7009f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" /></span> then <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 h:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d91a802daa0df93b0c6fcc9e2065fac31ce08e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.644ex; height:2.176ex;" alt="{\displaystyle h:X\to Y}" /></span> is a continuous linear map and the maps <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 h,h_{1},h_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>,</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h,h_{1},h_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af2d6e7240d9fff7bd8afedf5e26a93afe9b3e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.95ex; height:2.509ex;" alt="{\displaystyle h,h_{1},h_{2},\ldots }" /></span> are equicontinuous. </p> </div> <p>If in addition the domain is a <a href="/wiki/Banach_space" title="Banach space">Banach space</a> and the codomain is a <a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">normed space</a> then <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 \|h\|\leq \liminf _{n\to \infty }\left\|h_{n}\right\|<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>h</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <munder> <mo movablelimits="true" form="prefix">lim inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mo symmetric="true">‖</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|h\|\leq \liminf _{n\to \infty }\left\|h_{n}\right\|<\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e8658a471b207c7a76255892d4a94f6fe97ef8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.37ex; height:3.843ex;" alt="{\displaystyle \|h\|\leq \liminf _{n\to \infty }\left\|h_{n}\right\|<\infty .}" /></span> </p> <div class="mw-heading mw-heading4"><h4 id="Complete_metrizable_domain">Complete metrizable domain</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_boundedness_principle&action=edit&section=9" title="Edit section: Complete metrizable domain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="#CITEREFDieudonné1970">Dieudonné (1970)</a> proves a weaker form of this theorem with <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet spaces</a> rather than the usual Banach spaces. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem<sup id="cite_ref-FOOTNOTERudin199142−47_2-3" class="reference"><a href="#cite_note-FOOTNOTERudin199142−47-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></strong><span class="theoreme-tiret"> — </span>Let <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 H\subseteq L(X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>⊆</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\subseteq L(X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48336f24bfeb89b5868268e742d1d7ded8b37b24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.341ex; height:2.843ex;" alt="{\displaystyle H\subseteq L(X,Y)}" /></span> be a set of continuous linear operators from a <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">complete</a> <a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">metrizable topological vector space</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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}" /></span> (such as a <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet space</a> or an <a href="/wiki/F-space" title="F-space">F-space</a>) into a <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</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 Y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" /></span> If for every <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\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" /></span> the <a href="/wiki/Orbit_(group_theory)" class="mw-redirect" title="Orbit (group theory)">orbit</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x):=\{h(x):h\in H\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>h</mi> <mo>∈</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x):=\{h(x):h\in H\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5a48fe43e6006f982590bd2c3433846c2ef379" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.931ex; height:2.843ex;" alt="{\displaystyle H(x):=\{h(x):h\in H\}}" /></span> is a <a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">bounded subset</a> of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> then <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /></span> is equicontinuous. </p><p>So in particular, if <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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}" /></span> is also a <a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">normed space</a> and if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{h\in H}\|h(x)\|<\infty \quad {\text{ for every }}x\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>∈</mo> <mi>H</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> for every </mtext> </mrow> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{h\in H}\|h(x)\|<\infty \quad {\text{ for every }}x\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/063d7efd8f73de903185cdaa718e39a89d37e8b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.193ex; height:4.509ex;" alt="{\displaystyle \sup _{h\in H}\|h(x)\|<\infty \quad {\text{ for every }}x\in X,}" /></span> then <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /></span> is equicontinuous. </p> </div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_boundedness_principle&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled space</a> – Type of topological vector space</li> <li><a href="/wiki/Ursescu_theorem" title="Ursescu theorem">Ursescu theorem</a> – Generalization of closed graph, open mapping, and uniform boundedness theorem</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_boundedness_principle&action=edit&section=11" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> </div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_boundedness_principle&action=edit&section=12" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-FOOTNOTEShtern2001-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEShtern2001_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFShtern2001">Shtern 2001</a>.</span> </li> <li id="cite_note-FOOTNOTERudin199142−47-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTERudin199142−47_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTERudin199142−47_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTERudin199142−47_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-FOOTNOTERudin199142−47_2-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFRudin1991">Rudin 1991</a>, pp. 42−47.</span> </li> <li id="cite_note-FOOTNOTERudin199146-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTERudin199146_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTERudin199146_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTERudin199146_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFRudin1991">Rudin 1991</a>, p. 46.</span> </li> <li id="cite_note-FOOTNOTERudin199145−46-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERudin199145−46_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRudin1991">Rudin 1991</a>, pp. 45−46.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_boundedness_principle&action=edit&section=13" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output 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Retrieved <span class="nowrap">2020-07-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+des+Op%C3%A9rations+Lin%C3%A9aires&rft.place=Warszawa&rft.series=Monografie+Matematyczne&rft.pub=Subwencji+Funduszu+Kultury+Narodowej&rft.date=1932&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0005.20901%23id-name%3DZbl&rft.aulast=Banach&rft.aufirst=Stefan&rft_id=http%3A%2F%2Fkielich.amu.edu.pl%2FStefan_Banach%2Fpdf%2Fteoria-operacji-fr%2Fbanach-teorie-des-operations-lineaires.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniform+boundedness+principle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBourbaki1987" class="citation book cs1"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1987) [1981]. <i>Topological Vector Spaces: Chapters 1–5</i>. <a href="/wiki/%C3%89l%C3%A9ments_de_math%C3%A9matique" title="Éléments de mathématique">Éléments de mathématique</a>. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-13627-4" title="Special:BookSources/3-540-13627-4"><bdi>3-540-13627-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/17499190">17499190</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces%3A+Chapters+1%E2%80%935&rft.place=Berlin+New+York&rft.series=%C3%89l%C3%A9ments+de+math%C3%A9matique&rft.pub=Springer-Verlag&rft.date=1987&rft_id=info%3Aoclcnum%2F17499190&rft.isbn=3-540-13627-4&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniform+boundedness+principle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDieudonné1970" class="citation cs2"><a href="/wiki/Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean</a> (1970), <i>Treatise on analysis, Volume 2</i>, Academic Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Treatise+on+analysis%2C+Volume+2&rft.pub=Academic+Press&rft.date=1970&rft.aulast=Dieudonn%C3%A9&rft.aufirst=Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniform+boundedness+principle" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHusainKhaleelulla1978" class="citation book cs1">Husain, Taqdir; Khaleelulla, S. 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Boca Raton, FL: CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1584888666" title="Special:BookSources/978-1584888666"><bdi>978-1584888666</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/144216834">144216834</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces&rft.place=Boca+Raton%2C+FL&rft.series=Pure+and+applied+mathematics&rft.edition=Second&rft.pub=CRC+Press&rft.date=2011&rft_id=info%3Aoclcnum%2F144216834&rft.isbn=978-1584888666&rft.aulast=Narici&rft.aufirst=Lawrence&rft.au=Beckenstein%2C+Edward&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniform+boundedness+principle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRudin1966" class="citation cs2"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1966), <i>Real and complex analysis</i>, McGraw-Hill</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+and+complex+analysis&rft.pub=McGraw-Hill&rft.date=1966&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniform+boundedness+principle" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRudin1991" class="citation book cs1"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi"><i>Functional Analysis</i></a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="/wiki/McGraw-Hill_Science/Engineering/Math" class="mw-redirect" title="McGraw-Hill Science/Engineering/Math">McGraw-Hill Science/Engineering/Math</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-054236-5" title="Special:BookSources/978-0-07-054236-5"><bdi>978-0-07-054236-5</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/21163277">21163277</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Analysis&rft.place=New+York%2C+NY&rft.series=International+Series+in+Pure+and+Applied+Mathematics&rft.edition=Second&rft.pub=McGraw-Hill+Science%2FEngineering%2FMath&rft.date=1991&rft_id=info%3Aoclcnum%2F21163277&rft.isbn=978-0-07-054236-5&rft.aulast=Rudin&rft.aufirst=Walter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffunctionalanalys00rudi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniform+boundedness+principle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchaeferWolff1999" class="citation book cs1"><a href="/wiki/Helmut_H._Schaefer" title="Helmut H. Schaefer">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). <i>Topological Vector Spaces</i>. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-7155-0" title="Special:BookSources/978-1-4612-7155-0"><bdi>978-1-4612-7155-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/840278135">840278135</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces&rft.place=New+York%2C+NY&rft.series=GTM&rft.edition=Second&rft.pub=Springer+New+York+Imprint+Springer&rft.date=1999&rft_id=info%3Aoclcnum%2F840278135&rft.isbn=978-1-4612-7155-0&rft.aulast=Schaefer&rft.aufirst=Helmut+H.&rft.au=Wolff%2C+Manfred+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniform+boundedness+principle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchechter1996" class="citation book cs1"><a href="/wiki/Eric_Schechter" title="Eric Schechter">Schechter, Eric</a> (1996). <i>Handbook of Analysis and Its Foundations</i>. San Diego, CA: Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-622760-4" title="Special:BookSources/978-0-12-622760-4"><bdi>978-0-12-622760-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/175294365">175294365</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Analysis+and+Its+Foundations&rft.place=San+Diego%2C+CA&rft.pub=Academic+Press&rft.date=1996&rft_id=info%3Aoclcnum%2F175294365&rft.isbn=978-0-12-622760-4&rft.aulast=Schechter&rft.aufirst=Eric&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniform+boundedness+principle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFShtern2001" class="citation cs2">Shtern, A.I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Uniform_boundedness_principle">"Uniform boundedness principle"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Uniform+boundedness+principle&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Shtern&rft.aufirst=A.I.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DUniform_boundedness_principle&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniform+boundedness+principle" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSokal2011" class="citation cs2"><a href="/wiki/Alan_Sokal" title="Alan Sokal">Sokal, Alan</a> (2011), "A really simple elementary proof of the uniform boundedness theorem", <i><a href="/wiki/Amer._Math._Monthly" class="mw-redirect" title="Amer. Math. Monthly">Amer. Math. Monthly</a></i>, <b>118</b> (5): <span class="nowrap">450–</span>452, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1005.1585">1005.1585</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2Famer.math.monthly.118.05.450">10.4169/amer.math.monthly.118.05.450</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:41853641">41853641</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Amer.+Math.+Monthly&rft.atitle=A+really+simple+elementary+proof+of+the+uniform+boundedness+theorem&rft.volume=118&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E450-%3C%2Fspan%3E452&rft.date=2011&rft_id=info%3Aarxiv%2F1005.1585&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A41853641%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.4169%2Famer.math.monthly.118.05.450&rft.aulast=Sokal&rft.aufirst=Alan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniform+boundedness+principle" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTrèves2006" class="citation book cs1"><a href="/wiki/Fran%C3%A7ois_Tr%C3%A8ves" title="François Trèves">Trèves, François</a> (2006) [1967]. <i>Topological Vector Spaces, Distributions and Kernels</i>. Mineola, N.Y.: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45352-1" title="Special:BookSources/978-0-486-45352-1"><bdi>978-0-486-45352-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/853623322">853623322</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces%2C+Distributions+and+Kernels&rft.place=Mineola%2C+N.Y.&rft.pub=Dover+Publications&rft.date=2006&rft_id=info%3Aoclcnum%2F853623322&rft.isbn=978-0-486-45352-1&rft.aulast=Tr%C3%A8ves&rft.aufirst=Fran%C3%A7ois&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniform+boundedness+principle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWilansky2013" class="citation book cs1"><a href="/wiki/Albert_Wilansky" title="Albert Wilansky">Wilansky, Albert</a> (2013). <i>Modern Methods in Topological Vector Spaces</i>. Mineola, New York: Dover Publications, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-49353-4" title="Special:BookSources/978-0-486-49353-4"><bdi>978-0-486-49353-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/849801114">849801114</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Methods+in+Topological+Vector+Spaces&rft.place=Mineola%2C+New+York&rft.pub=Dover+Publications%2C+Inc&rft.date=2013&rft_id=info%3Aoclcnum%2F849801114&rft.isbn=978-0-486-49353-4&rft.aulast=Wilansky&rft.aufirst=Albert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniform+boundedness+principle" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output 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abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Functional_analysis" title="Template:Functional analysis"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Functional_analysis" title="Template talk:Functional analysis"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Functional_analysis" title="Special:EditPage/Template:Functional analysis"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Functional_analysis_(topics_–_glossary)364" style="font-size:114%;margin:0 4em"><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> (<a href="/wiki/List_of_functional_analysis_topics" title="List of functional analysis topics">topics</a> – <a href="/wiki/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="/wiki/Banach_space" title="Banach space">Banach</a></li> <li><a href="/wiki/Besov_space" title="Besov space">Besov</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a></li> <li><a href="/wiki/H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder</a></li> <li><a href="/wiki/Nuclear_space" title="Nuclear space">Nuclear</a></li> <li><a href="/wiki/Orlicz_space" title="Orlicz space">Orlicz</a></li> <li><a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev</a></li> <li><a href="/wiki/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="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual</a> (<a href="/wiki/Dual_space#Algebraic_dual_space" title="Dual space">Algebraic</a> / <a href="/wiki/Dual_space#Continuous_dual_space" title="Dual space">Topological</a>)</li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a></li> <li><a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/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%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Theorems</a></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="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li> <li><a href="/wiki/Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph</a></li> <li><a class="mw-selflink selflink">Uniform boundedness principle</a></li> <li><a href="/wiki/Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li> <li><a href="/wiki/Min-max_theorem" title="Min-max theorem">Min–max</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li> <li><a href="/wiki/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="/wiki/Adjoint_operator" class="mw-redirect" title="Adjoint operator">Adjoint</a></li> <li><a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Nuclear_operator" title="Nuclear operator">Nuclear</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li> <li><a href="/wiki/Unbounded_operator" title="Unbounded operator">Unbounded</a></li> <li><a href="/wiki/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="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a></li> <li><a href="/wiki/Mahler%27s_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="/wiki/Hardy_space" title="Hardy space">Hardy space</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li> <li><a href="/wiki/Heat_kernel" title="Heat kernel">Heat kernel</a></li> <li><a href="/wiki/Index_theorem" class="mw-redirect" title="Index theorem">Index theorem</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li> <li><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a></li> <li><a href="/wiki/Integral_linear_operator" title="Integral linear operator">Integral linear operator</a></li> <li><a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li> <li><a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li> <li><a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a> (or <a href="/wiki/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="/wiki/Approximation_property" title="Approximation property">Approximation property</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced set</a></li> <li><a href="/wiki/Choquet_theory" title="Choquet theory">Choquet theory</a></li> <li><a href="/wiki/Weak_topology" title="Weak topology">Weak topology</a></li> <li><a href="/wiki/Banach%E2%80%93Mazur_distance" class="mw-redirect" title="Banach–Mazur distance">Banach–Mazur distance</a></li> <li><a href="/wiki/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"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Functional_analysis" title="Category:Functional analysis">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Topological_vector_spaces" title="Template:Topological vector spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Topological_vector_spaces" title="Template talk:Topological vector spaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Topological_vector_spaces" title="Special:EditPage/Template:Topological vector spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Topological_vector_spaces_(TVSs)267" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/Banach_space" title="Banach space">Banach space</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">Completeness</a></li> <li><a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">Continuous linear operator</a></li> <li><a href="/wiki/Linear_form" title="Linear form">Linear functional</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet space</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex space</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizability</a></li> <li><a href="/wiki/Operator_topologies" title="Operator topologies">Operator topologies</a></li> <li><a href="/wiki/Topological_vector_space" title="Topological vector space">Topological vector space</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Main results</a></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="/wiki/Anderson%E2%80%93Kadec_theorem" title="Anderson–Kadec theorem">Anderson–Kadec</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li> <li><a href="/wiki/Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph theorem</a></li> <li><a href="/wiki/F._Riesz%27s_theorem" title="F. Riesz's theorem">F. Riesz's</a></li> <li><a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a> (<a href="/wiki/Hyperplane_separation_theorem" title="Hyperplane separation theorem">hyperplane separation</a></li> <li><a href="/wiki/Vector-valued_Hahn%E2%80%93Banach_theorems" title="Vector-valued Hahn–Banach theorems">Vector-valued Hahn–Banach</a>)</li> <li><a href="/wiki/Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">Open mapping (Banach–Schauder)</a> <ul><li><a href="/wiki/Bounded_inverse_theorem" class="mw-redirect" title="Bounded inverse theorem">Bounded inverse</a></li></ul></li> <li><a class="mw-selflink selflink">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="/wiki/Bilinear_operator" class="mw-redirect" title="Bilinear operator">Bilinear operator</a> <ul><li><a href="/wiki/Bilinear_form" title="Bilinear form">form</a></li></ul></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a> <ul><li><a href="/wiki/Almost_open_linear_map" class="mw-redirect" title="Almost open linear map">Almost open</a></li> <li><a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">Continuous</a></li> <li><a href="/wiki/Closed_linear_operator" title="Closed linear operator">Closed</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a></li> <li><a href="/wiki/Densely_defined_operator" title="Densely defined operator">Densely defined</a></li> <li><a href="/wiki/Discontinuous_linear_map" title="Discontinuous linear map">Discontinuous</a></li></ul></li> <li><a href="/wiki/Topological_homomorphism" title="Topological homomorphism">Topological homomorphism</a></li> <li><a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">Functional</a> <ul><li><a href="/wiki/Linear_form" title="Linear form">Linear</a></li> <li><a href="/wiki/Bilinear_form" title="Bilinear form">Bilinear</a></li> <li><a href="/wiki/Sesquilinear_form" title="Sesquilinear form">Sesquilinear</a></li></ul></li> <li><a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">Norm</a></li> <li><a href="/wiki/Seminorm" title="Seminorm">Seminorm</a></li> <li><a href="/wiki/Sublinear_function" title="Sublinear function">Sublinear function</a></li> <li><a href="/wiki/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="/wiki/Absolutely_convex_set" title="Absolutely convex set">Absolutely convex/disk</a></li> <li><a href="/wiki/Absorbing_set" title="Absorbing set">Absorbing/Radial</a></li> <li><a href="/wiki/Affine_space" title="Affine space">Affine</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced/Circled</a></li> <li><a href="/wiki/Auxiliary_normed_space" title="Auxiliary normed space">Banach disks</a></li> <li><a href="/wiki/Bounding_point" title="Bounding point">Bounding points</a></li> <li><a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Bounded</a></li> <li><a href="/wiki/Complemented_subspace" title="Complemented subspace">Complemented subspace</a></li> <li><a href="/wiki/Convex_set" title="Convex set">Convex</a></li> <li><a href="/wiki/Convex_cone" title="Convex cone">Convex cone <span style="font-size:85%;">(subset)</span></a></li> <li><a href="/wiki/Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">Linear cone <span style="font-size:85%;">(subset)</span></a></li> <li><a href="/wiki/Extreme_point" title="Extreme point">Extreme point</a></li> <li><a href="/wiki/Totally_bounded_space#Topological_vector_spaces" title="Totally bounded space">Pre-compact/Totally bounded</a></li> <li><a href="/wiki/Prevalent_and_shy_sets" title="Prevalent and shy sets">Prevalent/Shy</a></li> <li><a href="/wiki/Radial_set" title="Radial set">Radial</a></li> <li><a href="/wiki/Star_domain" title="Star domain">Radially convex/Star-shaped</a></li> <li><a href="/wiki/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="/wiki/Affine_hull" title="Affine hull">Affine hull</a></li> <li>(<a href="/wiki/Algebraic_interior#Relative_algebraic_interior" title="Algebraic interior">Relative</a>) <a href="/wiki/Algebraic_interior" title="Algebraic interior">Algebraic interior (core)</a></li> <li><a href="/wiki/Convex_hull" title="Convex hull">Convex hull</a></li> <li><a href="/wiki/Linear_span" title="Linear span">Linear span</a></li> <li><a href="/wiki/Minkowski_addition" title="Minkowski addition">Minkowski addition</a></li> <li><a href="/wiki/Polar_set" title="Polar set">Polar</a></li> <li>(<a href="/wiki/Algebraic_interior#Quasi_relative_interior" title="Algebraic interior">Quasi</a>) <a href="/wiki/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="/wiki/Asplund_space" title="Asplund space">Asplund</a></li> <li><a href="/wiki/Ptak_space" title="Ptak space">B-complete/Ptak</a></li> <li><a href="/wiki/Banach_space" title="Banach space">Banach</a></li> <li>(<a href="/wiki/Countably_barrelled_space" title="Countably barrelled space">Countably</a>) <a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/BK-space" title="BK-space">BK-space</a></li> <li>(<a href="/wiki/Ultrabornological_space" title="Ultrabornological space">Ultra-</a>) <a href="/wiki/Bornological_space" title="Bornological space">Bornological</a></li> <li><a href="/wiki/Brauner_space" title="Brauner space">Brauner</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/Convenient_vector_space" title="Convenient vector space">Convenient</a></li> <li><a href="/wiki/DF-space" title="DF-space">(DF)-space</a></li> <li><a href="/wiki/Distinguished_space" title="Distinguished space">Distinguished</a></li> <li><a href="/wiki/F-space" title="F-space">F-space</a></li> <li><a href="/wiki/FK-AK_space" title="FK-AK space">FK-AK space</a></li> <li><a href="/wiki/FK-space" title="FK-space">FK-space</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a> <ul><li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces#Tame_Fréchet_spaces" title="Differentiation in Fréchet spaces">tame Fréchet</a></li></ul></li> <li><a href="/wiki/Grothendieck_space" title="Grothendieck space">Grothendieck</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a></li> <li><a href="/wiki/Infrabarreled_space" class="mw-redirect" title="Infrabarreled space">Infrabarreled</a></li> <li><a href="/wiki/Interpolation_space" title="Interpolation space">Interpolation space</a></li> <li><a href="/wiki/K-space_(functional_analysis)" title="K-space (functional analysis)">K-space</a></li> <li><a href="/wiki/LB-space" title="LB-space">LB-space</a></li> <li><a href="/wiki/LF-space" title="LF-space">LF-space</a></li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex space</a></li> <li><a href="/wiki/Mackey_space" title="Mackey space">Mackey</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">(Pseudo)Metrizable</a></li> <li><a href="/wiki/Montel_space" title="Montel space">Montel</a></li> <li><a href="/wiki/Quasibarrelled_space" class="mw-redirect" title="Quasibarrelled space">Quasibarrelled</a></li> <li><a href="/wiki/Quasi-complete" class="mw-redirect" title="Quasi-complete">Quasi-complete</a></li> <li><a href="/wiki/Quasinorm" title="Quasinorm">Quasinormed</a></li> <li>(<a href="/wiki/Polynomially_reflexive_space" title="Polynomially reflexive space">Polynomially</a></li> <li><a href="/wiki/Semi-reflexive_space" title="Semi-reflexive space">Semi-</a>) <a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Riesz_space" title="Riesz space">Riesz</a></li> <li><a href="/wiki/Schwartz_TVS" class="mw-redirect" title="Schwartz TVS">Schwartz</a></li> <li><a href="/wiki/Semi-complete" class="mw-redirect" title="Semi-complete">Semi-complete</a></li> <li><a href="/wiki/Smith_space" title="Smith space">Smith</a></li> <li><a href="/wiki/Stereotype_space" class="mw-redirect" title="Stereotype space">Stereotype</a></li> <li>(<a href="/wiki/B-convex_space" title="B-convex space">B</a></li> <li><a href="/wiki/Strictly_convex_space" title="Strictly convex space">Strictly</a></li> <li><a href="/wiki/Uniformly_convex_space" title="Uniformly convex space">Uniformly</a>) convex</li> <li>(<a href="/wiki/Quasi-ultrabarrelled_space" title="Quasi-ultrabarrelled space">Quasi-</a>) <a href="/wiki/Ultrabarrelled_space" title="Ultrabarrelled space">Ultrabarrelled</a></li> <li><a href="/wiki/Uniformly_smooth_space" title="Uniformly smooth space">Uniformly smooth</a></li> <li><a href="/wiki/Webbed_space" title="Webbed space">Webbed</a></li> <li><a href="/wiki/Approximation_property" title="Approximation property">With the approximation property</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Topological_vector_spaces" title="Category:Topological vector spaces">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Boundedness_and_bornology95" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Boundedness_and_bornology" title="Template:Boundedness and bornology"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Boundedness_and_bornology" title="Template talk:Boundedness and bornology"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Boundedness_and_bornology" title="Special:EditPage/Template:Boundedness and bornology"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Boundedness_and_bornology95" style="font-size:114%;margin:0 4em"><a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Boundedness</a> and <a href="/wiki/Bornology" title="Bornology">bornology</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled space</a></li> <li><a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Bounded set</a></li> <li><a href="/wiki/Bornological_space" title="Bornological space">Bornological space</a></li> <li>(<a href="/wiki/Vector_bornology" title="Vector bornology">Vector</a>) <a href="/wiki/Bornology" title="Bornology">Bornology</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="/wiki/Unbounded_operator" title="Unbounded operator">Un</a>)<a href="/wiki/Bounded_operator" title="Bounded operator">Bounded operator</a></li> <li><a class="mw-selflink selflink">Uniform boundedness principle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Subsets</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrelled_set" title="Barrelled set">Barrelled set</a></li> <li><a href="/wiki/Bornivorous_set" title="Bornivorous set">Bornivorous set</a></li> <li><a href="/wiki/Saturated_family" title="Saturated family">Saturated family</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related spaces</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="/wiki/Countably_barrelled_space" title="Countably barrelled space">Countably</a>) <a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled space</a></li> <li>(<a href="/wiki/Countably_quasi-barrelled_space" title="Countably quasi-barrelled space">Countably</a>) <a href="/wiki/Quasi-barrelled_space" class="mw-redirect" title="Quasi-barrelled space">Quasi-barrelled space</a></li> <li><a href="/wiki/Infrabarrelled_space" title="Infrabarrelled space">Infrabarrelled space</a></li> <li>(<a href="/wiki/Quasi-ultrabarrelled_space" title="Quasi-ultrabarrelled space">Quasi-</a>) <a href="/wiki/Ultrabarrelled_space" title="Ultrabarrelled space">Ultrabarrelled space</a></li> <li><a href="/wiki/Ultrabornological_space" title="Ultrabornological 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Uniform boundedness principle - Wikipedia