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href="#Weaker_structures_inducing_the_topology_of_uniform_convergence"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Weaker structures inducing the topology of uniform convergence</span> </div> </a> <button aria-controls="toc-Weaker_structures_inducing_the_topology_of_uniform_convergence-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Weaker structures inducing the topology of uniform convergence subsection</span> </button> <ul id="toc-Weaker_structures_inducing_the_topology_of_uniform_convergence-sublist" class="vector-toc-list"> <li id="toc-Uniform_metric" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniform_metric"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Uniform metric</span> </div> </a> <ul id="toc-Uniform_metric-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uniformity_of_uniform_convergence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniformity_of_uniform_convergence"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Uniformity of uniform convergence</span> </div> </a> <ul id="toc-Uniformity_of_uniform_convergence-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul 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class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Norma_uniforme" title="Norma uniforme – Italian" lang="it" hreflang="it" data-title="Norma uniforme" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%80%E6%A7%98%E3%83%8E%E3%83%AB%E3%83%A0" title="一様ノルム – Japanese" lang="ja" hreflang="ja" data-title="一様ノルム" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Norma_do_supremo" title="Norma do supremo – Portuguese" lang="pt" hreflang="pt" data-title="Norma do supremo" data-language-autonym="Português" data-language-local-name="Portuguese" 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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">This article is about the function space norm. For the finite-dimensional vector space distance, see <a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a>. For the uniformity norm in additive combinatorics, see <a href="/wiki/Gowers_norm" title="Gowers norm">Gowers norm</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Uniform_norm" title="Special:EditPage/Uniform norm">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Uniform+norm%22">"Uniform norm"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Uniform+norm%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Uniform+norm%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Uniform+norm%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Uniform+norm%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Uniform+norm%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">December 2009</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Vector_norm_sup.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Vector_norm_sup.svg/144px-Vector_norm_sup.svg.png" decoding="async" width="144" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Vector_norm_sup.svg/216px-Vector_norm_sup.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Vector_norm_sup.svg/288px-Vector_norm_sup.svg.png 2x" data-file-width="144" data-file-height="145" /></a><figcaption>The perimeter of the square is the set of points in <span class="texhtml">ℝ<sup>2</sup></span> where the sup norm equals a fixed positive constant. For example, points <span class="texhtml">(2, 0)</span>, <span class="texhtml">(2, 1)</span>, and <span class="texhtml">(2, 2)</span> lie along the perimeter of a square and belong to the set of vectors whose sup norm is 2.</figcaption></figure> <p>In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the <b>uniform norm</b> (or <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="sup_norm"></span><span class="vanchor-text">sup norm</span></span></b>) assigns to <a href="/wiki/Real_number" title="Real number">real-</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a>-valued <a href="/wiki/Bounded_function" title="Bounded function">bounded functions</a> <span class="nowrap">⁠<span class="mwe-math-element"><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>⁠</span> defined on a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <span class="nowrap">⁠<span class="mwe-math-element"><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>⁠</span> the non-negative number </p> <dl><dd><span class="mwe-math-element"><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\|_{\infty }=\|f\|_{\infty ,S}=\sup \left\{\,|f(s)|:s\in S\,\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mrow> <mo>{</mo> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mspace width="thinmathspace" /> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }=\|f\|_{\infty ,S}=\sup \left\{\,|f(s)|:s\in S\,\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a46e7e33db76a8ff8f8fc435548f0d73d878725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.533ex; height:3.009ex;" alt="{\displaystyle \|f\|_{\infty }=\|f\|_{\infty ,S}=\sup \left\{\,|f(s)|:s\in S\,\right\}.}"></span></dd></dl> <p>This <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> is also called the <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="supremum_norm"></span><span class="vanchor-text">supremum norm</span></span>,</b> the <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Chebyshev_norm"></span><span class="vanchor-text">Chebyshev norm</span></span>,</b> the <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="infinity_norm"></span><span class="vanchor-text">infinity norm</span></span>,</b> or, when the <a href="/wiki/Infimum_and_supremum" title="Infimum and supremum">supremum</a> is in fact the maximum, the <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="max_norm"></span><span class="vanchor-text">max norm</span></span></b>. The name "uniform norm" derives from the fact that a sequence of functions <span class="nowrap">⁠<span class="mwe-math-element"><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\{f_{n}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{f_{n}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/758c0eee277b73e394f51afb221728c840677dfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.683ex; height:2.843ex;" alt="{\displaystyle \left\{f_{n}\right\}}"></span>⁠</span> converges to <span class="nowrap">⁠<span class="mwe-math-element"><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>⁠</span> under the <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> derived from the uniform norm <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <span class="nowrap">⁠<span class="mwe-math-element"><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/b2702450f0458a5e01a698e248af552a7fab2b50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.509ex;" alt="{\displaystyle f_{n}}"></span>⁠</span> converges to <span class="nowrap">⁠<span class="mwe-math-element"><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>⁠</span> <a href="/wiki/Uniform_convergence" title="Uniform convergence">uniformly</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>If <span class="nowrap">⁠<span class="mwe-math-element"><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>⁠</span> is a <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a> on a <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">closed and bounded interval</a>, or more generally a <a href="/wiki/Compact_space" title="Compact space">compact</a> set, then it is bounded and the <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a> in the above definition is attained by the Weierstrass <a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">extreme value theorem</a>, so we can replace the supremum by the maximum. In this case, the norm is also called the <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="maximum_norm"></span><span class="vanchor-text">maximum norm</span></span></b>. In particular, if <span class="nowrap">⁠<span class="mwe-math-element"><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>⁠</span> is some vector 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=\left(x_{1},x_{2},\ldots ,x_{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\left(x_{1},x_{2},\ldots ,x_{n}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6db41d0f18aee38b5dd28c3ba9a20f0b39117422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.766ex; height:2.843ex;" alt="{\displaystyle x=\left(x_{1},x_{2},\ldots ,x_{n}\right)}"></span> in <a href="/wiki/Finite_set" title="Finite set">finite</a> dimensional <a href="/wiki/Coordinate_space" class="mw-redirect" title="Coordinate space">coordinate space</a>, it takes the form: </p> <dl><dd><span class="mwe-math-element"><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\|_{\infty }:=\max \left(\left|x_{1}\right|,\ldots ,\left|x_{n}\right|\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>:=</mo> <mo movablelimits="true" form="prefix">max</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>|</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow> <mo>|</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{\infty }:=\max \left(\left|x_{1}\right|,\ldots ,\left|x_{n}\right|\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9448bc2d9e21c860bf05b167dc9064c80dfe98d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.916ex; height:2.843ex;" alt="{\displaystyle \|x\|_{\infty }:=\max \left(\left|x_{1}\right|,\ldots ,\left|x_{n}\right|\right).}"></span></dd></dl> <p>This is called the <a href="/wiki/L-infinity" title="L-infinity"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8348195cf09473662c6f59e6717722a6fc01d0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}"></span>-norm</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_norm&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uniform norms are defined, in general, for <a href="/wiki/Bounded_function" title="Bounded function">bounded functions</a> valued in a <a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">normed space</a>. 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 set 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 (Y,\|\|_{Y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Y,\|\|_{Y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ed0a7fe9d1c4ef82c2a9c70b5727f2497554fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.428ex; height:2.843ex;" alt="{\displaystyle (Y,\|\|_{Y})}"></span> be a <a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">normed space</a>. On 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 Y^{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y^{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233b4c76e7ea7aad9de5488491e1c6c7363c0bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.533ex; height:2.509ex;" alt="{\displaystyle Y^{X}}"></span> of functions 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>, there is an <a href="/w/index.php?title=Extended_norm&action=edit&redlink=1" class="new" title="Extended norm (page does not exist)">extended norm</a> defined by </p> <dl><dd><span class="mwe-math-element"><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\|=\sup _{x\in X}\|f(x)\|_{Y}\in [0,\infty ].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>f</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> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|=\sup _{x\in X}\|f(x)\|_{Y}\in [0,\infty ].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf1c685cb8c7be23325638f8da4bb55f93707107" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.12ex; height:4.509ex;" alt="{\displaystyle \|f\|=\sup _{x\in X}\|f(x)\|_{Y}\in [0,\infty ].}"></span></dd></dl> <p>This is in general an extended norm since the 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> may not be bounded. Restricting this extended norm to the bounded functions (i.e., the functions with finite above extended norm) yields a (finite-valued) norm, called the <b>uniform norm</b> 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 Y^{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y^{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233b4c76e7ea7aad9de5488491e1c6c7363c0bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.533ex; height:2.509ex;" alt="{\displaystyle Y^{X}}"></span>. Note that the definition of uniform norm does not rely on any additional structure on 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 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>, although in practice <span class="mwe-math-element"><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 often at least a <a href="/wiki/Topological_space" title="Topological space">topological space</a>. </p><p>The convergence 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 Y^{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y^{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233b4c76e7ea7aad9de5488491e1c6c7363c0bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.533ex; height:2.509ex;" alt="{\displaystyle Y^{X}}"></span> in the topology induced by the uniform extended norm is the <a href="/wiki/Uniform_convergence" title="Uniform convergence">uniform convergence</a>, for sequences, and also for <a href="/wiki/Net_(mathematics)" title="Net (mathematics)">nets</a> and <a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">filters</a> 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 Y^{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y^{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233b4c76e7ea7aad9de5488491e1c6c7363c0bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.533ex; height:2.509ex;" alt="{\displaystyle Y^{X}}"></span>. </p><p>We can define closed sets and closures of sets with respect to this metric topology; closed sets in the uniform norm are sometimes called <i>uniformly closed</i> and closures <i>uniform closures</i>. The uniform closure of a set of functions A is the space of all functions that can be approximated by a sequence of uniformly-converging 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 A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a71bf21ad35b8fe05555041d54d1e17eeb0f490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.39ex; height:2.176ex;" alt="{\displaystyle A.}"></span> For instance, one restatement of the <a href="/wiki/Stone%E2%80%93Weierstrass_theorem" title="Stone–Weierstrass theorem">Stone–Weierstrass theorem</a> is that the set of all 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 [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> is the uniform closure of the set of polynomials 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 [a,b].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba5cb29655f824ce80a0b6a32d9326d0e8742cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.202ex; height:2.843ex;" alt="{\displaystyle [a,b].}"></span> </p><p>For complex <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> functions over a compact space, this turns it into a <a href="/wiki/C-star_algebra" class="mw-redirect" title="C-star algebra">C* algebra</a> (cf. <a href="/wiki/Gelfand_representation" title="Gelfand representation">Gelfand representation</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Weaker_structures_inducing_the_topology_of_uniform_convergence">Weaker structures inducing the topology of uniform convergence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_norm&action=edit&section=2" title="Edit section: Weaker structures inducing the topology of uniform convergence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Uniform_metric">Uniform metric</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_norm&action=edit&section=3" title="Edit section: Uniform metric"><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/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a></div> <p>The <b>uniform metric</b> between two bounded functions <span class="mwe-math-element"><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,g\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f99c3354a919e8e246796bdc329e13c6604e95a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.83ex; height:2.509ex;" alt="{\displaystyle f,g\colon X\to Y}"></span> from a 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}"> <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 a <a href="/wiki/Metric_space" title="Metric space">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 (Y,d_{Y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Y,d_{Y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bce9785740aaefa247497db61b92d4a58720b9f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.312ex; height:2.843ex;" alt="{\displaystyle (Y,d_{Y})}"></span> is defined by </p> <dl><dd><span class="mwe-math-element"><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(f,g)=\sup _{x\in X}d_{Y}(f(x),g(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </munder> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(f,g)=\sup _{x\in X}d_{Y}(f(x),g(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458f10230732d03f19429cea5b771eb298b234b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.651ex; height:4.509ex;" alt="{\displaystyle d(f,g)=\sup _{x\in X}d_{Y}(f(x),g(x))}"></span></dd></dl> <p>The uniform metric is also called the <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Chebyshev_metric"></span><span class="vanchor-text">Chebyshev metric</span></span></b>, after <a href="/wiki/Pafnuty_Chebyshev" title="Pafnuty Chebyshev">Pafnuty Chebyshev</a>, who was first to systematically study it. In this case, <span class="mwe-math-element"><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> is bounded precisely 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 d(f,g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(f,g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197b0c1edd252f645fa5cad5945bebad3a8cbfa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.454ex; height:2.843ex;" alt="{\displaystyle d(f,g)}"></span> is finite for some <a href="/wiki/Constant_function" title="Constant function">constant function</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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>. If we allow unbounded functions, this formula does not yield a norm or metric in a strict sense, although the obtained so-called <a href="/wiki/Metric_(mathematics)#Generalized_metrics" class="mw-redirect" title="Metric (mathematics)">extended metric</a> still allows one to define a topology on the function space in question; the convergence is then still the <a href="/wiki/Uniform_convergence" title="Uniform convergence">uniform convergence</a>. In particular, a sequence <span class="mwe-math-element"><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\{f_{n}:n=1,2,3,\ldots \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{f_{n}:n=1,2,3,\ldots \right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/926126da81ce446bfae84831a42f8ee68130b48c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.425ex; height:2.843ex;" alt="{\displaystyle \left\{f_{n}:n=1,2,3,\ldots \right\}}"></span> <a href="/wiki/Uniform_convergence" title="Uniform convergence">converges uniformly</a> to a 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> 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 \lim _{n\rightarrow \infty }d(f_{n},f)=0.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty }d(f_{n},f)=0.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d25a470173253a414b50c8ba47f380f082e498b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.65ex; height:3.843ex;" alt="{\displaystyle \lim _{n\rightarrow \infty }d(f_{n},f)=0.\,}"></span> </p><p>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,\|\|_{Y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Y,\|\|_{Y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ed0a7fe9d1c4ef82c2a9c70b5727f2497554fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.428ex; height:2.843ex;" alt="{\displaystyle (Y,\|\|_{Y})}"></span> is a <a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">normed space</a>, then it is a <a href="/wiki/Metric_space" title="Metric space">metric space</a> in a natural way. The extended metric 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 Y^{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y^{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233b4c76e7ea7aad9de5488491e1c6c7363c0bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.533ex; height:2.509ex;" alt="{\displaystyle Y^{X}}"></span> induced by the uniform extended norm is the same as the uniform extended metric </p> <dl><dd><span class="mwe-math-element"><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(f,g)=\sup _{x\in X}\|f(x)-g(x)\|_{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</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 d(f,g)=\sup _{x\in X}\|f(x)-g(x)\|_{Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/650ada5ac88781dadf4e7e783d2ec0b485fdd98c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.765ex; height:4.509ex;" alt="{\displaystyle d(f,g)=\sup _{x\in X}\|f(x)-g(x)\|_{Y}}"></span></dd></dl> <p>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 Y^{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y^{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233b4c76e7ea7aad9de5488491e1c6c7363c0bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.533ex; height:2.509ex;" alt="{\displaystyle Y^{X}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Uniformity_of_uniform_convergence">Uniformity of uniform convergence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_norm&action=edit&section=4" title="Edit section: Uniformity of uniform convergence"><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">See also: <a href="/wiki/Topologies_on_spaces_of_linear_maps" title="Topologies on spaces of linear maps">Topologies on spaces of linear maps</a></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 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 set 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 (Y,{\mathcal {E}}_{Y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Y,{\mathcal {E}}_{Y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dab8c8e414106830ee8f0f020c0ce35e384ab7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.33ex; height:2.843ex;" alt="{\displaystyle (Y,{\mathcal {E}}_{Y})}"></span> be a <a href="/wiki/Uniform_space" title="Uniform space">uniform space</a>. A sequence <span class="mwe-math-element"><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"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </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/a32c571f35fd41f5ef78fecc130ac39cd2108c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.167ex; height:2.843ex;" alt="{\displaystyle (f_{n})}"></span> of functions 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 said to converge uniformly to a 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> if for each entourage <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\in {\mathcal {E}}_{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\in {\mathcal {E}}_{Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7137a3707ab510f03f3241bf3df35ba7f56f45a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.33ex; height:2.509ex;" alt="{\displaystyle E\in {\mathcal {E}}_{Y}}"></span> there is a natural number <span class="mwe-math-element"><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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63584d203ecb012a7bcb90f422408bbfe4018956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{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="{\displaystyle (f_{n}(x),f(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</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>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{n}(x),f(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a503a1d60d3cc882bce27a3b4a841213c2720dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.757ex; height:2.843ex;" alt="{\displaystyle (f_{n}(x),f(x))}"></span> belongs 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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> 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\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> 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 n\geq n_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq n_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff23d9c13da1b2a7ec6f7fab8fa413b129c6b6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.942ex; height:2.343ex;" alt="{\displaystyle n\geq n_{0}}"></span>. Similarly for a net. This is a convergence in a topology 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 Y^{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y^{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233b4c76e7ea7aad9de5488491e1c6c7363c0bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.533ex; height:2.509ex;" alt="{\displaystyle Y^{X}}"></span>. In fact, the sets </p> <dl><dd><span class="mwe-math-element"><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,g)\colon \forall x\in X\colon (f(x),g(x))\in E\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>E</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(f,g)\colon \forall x\in X\colon (f(x),g(x))\in E\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff16d16943358d74457a7699da44ec967ac1edb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.206ex; height:2.843ex;" alt="{\displaystyle \{(f,g)\colon \forall x\in X\colon (f(x),g(x))\in E\}}"></span></dd></dl> <p>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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> runs through entourages 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> form a fundamental system of entourages of a uniformity 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 Y^{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y^{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233b4c76e7ea7aad9de5488491e1c6c7363c0bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.533ex; height:2.509ex;" alt="{\displaystyle Y^{X}}"></span>, called the <b>uniformity of uniform convergence</b> 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 Y^{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y^{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233b4c76e7ea7aad9de5488491e1c6c7363c0bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.533ex; height:2.509ex;" alt="{\displaystyle Y^{X}}"></span>. The uniform convergence is precisely the convergence under its uniform topology. </p><p>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,d_{Y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Y,d_{Y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bce9785740aaefa247497db61b92d4a58720b9f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.312ex; height:2.843ex;" alt="{\displaystyle (Y,d_{Y})}"></span> is a <a href="/wiki/Metric_space" title="Metric space">metric space</a>, then it is by default equipped with the <a href="/w/index.php?title=Metric_uniformity&action=edit&redlink=1" class="new" title="Metric uniformity (page does not exist)">metric uniformity</a>. The metric uniformity 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 Y^{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y^{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233b4c76e7ea7aad9de5488491e1c6c7363c0bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.533ex; height:2.509ex;" alt="{\displaystyle Y^{X}}"></span> with respect to the uniform extended metric is then the uniformity of uniform convergence 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 Y^{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y^{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233b4c76e7ea7aad9de5488491e1c6c7363c0bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.533ex; height:2.509ex;" alt="{\displaystyle Y^{X}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_norm&action=edit&section=5" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The set of vectors whose infinity norm is a given constant, <span class="mwe-math-element"><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> <mo>,</mo> </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/ae5e8f9eb465084d3a00a24026b80652b74ef58e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.654ex; height:2.009ex;" alt="{\displaystyle c,}"></span> forms the surface of a <a href="/wiki/Hypercube" title="Hypercube">hypercube</a> with edge length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2c.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bfa91cd5d02a11f92ab00ec1266208ea773fef8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.816ex; height:2.176ex;" alt="{\displaystyle 2c.}"></span> </p><p>The reason for the subscript “<span class="mwe-math-element"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span>” is that 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 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> is continuous 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 \Vert f\Vert _{p}<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Vert f\Vert _{p}<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c92d737b611bcdbed2985d6c4dd62a0a6c8088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.085ex; height:3.009ex;" alt="{\displaystyle \Vert f\Vert _{p}<\infty }"></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 p\in (0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <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 p\in (0,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac7bcbe4e5dbf912889e0c2dc226b6782913c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.429ex; height:2.843ex;" alt="{\displaystyle p\in (0,\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 \lim _{p\to \infty }\|f\|_{p}=\|f\|_{\infty },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{p\to \infty }\|f\|_{p}=\|f\|_{\infty },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d8f167f42bad181792d22a1b52fb422a1bd5006" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; margin-left: -0.063ex; width:18.45ex; height:4.176ex;" alt="{\displaystyle \lim _{p\to \infty }\|f\|_{p}=\|f\|_{\infty },}"></span> where <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 \|f\|_{p}=\left(\int _{D}|f|^{p}\,d\mu \right)^{1/p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}=\left(\int _{D}|f|^{p}\,d\mu \right)^{1/p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2284e82a24bb3a53519ec62dbd6019b591998be" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.794ex; height:6.676ex;" alt="{\displaystyle \|f\|_{p}=\left(\int _{D}|f|^{p}\,d\mu \right)^{1/p}}"></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}"> <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> is the domain 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 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>; the integral amounts to a sum 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 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> is a <a href="/wiki/Discrete_set" class="mw-redirect" title="Discrete set">discrete set</a> (see <a href="/wiki/Norm_(mathematics)#p-norm" title="Norm (mathematics)"><i>p</i>-norm</a>). </p> <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_norm&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/L-infinity" title="L-infinity">L-infinity</a> – Space of bounded sequences</li> <li><a href="/wiki/Uniform_continuity" title="Uniform continuity">Uniform continuity</a> – Uniform restraint of the change in functions</li> <li><a href="/wiki/Uniform_space" title="Uniform space">Uniform space</a> – Topological space with a notion of uniform properties</li> <li><a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a> – Mathematical metric</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniform_norm&action=edit&section=7" title="Edit section: References"><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 class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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 .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFRudin1964" class="citation book cs1">Rudin, Walter (1964). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/principlesofmath00rudi"><i>Principles of Mathematical Analysis</i></a></span>. New York: McGraw-Hill. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/principlesofmath00rudi/page/151">151</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-054235-X" title="Special:BookSources/0-07-054235-X"><bdi>0-07-054235-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Mathematical+Analysis&rft.place=New+York&rft.pages=151&rft.pub=McGraw-Hill&rft.date=1964&rft.isbn=0-07-054235-X&rft.aulast=Rudin&rft.aufirst=Walter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprinciplesofmath00rudi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniform+norm" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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spaces</a></li> <li><a href="/wiki/Lp_space" title="Lp space"><i>L</i><sup><i>p</i></sup> spaces</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure</a> <ul><li><a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue</a></li></ul></li> <li><a href="/wiki/Measure_space" title="Measure space">Measure space</a></li> <li><a href="/wiki/Measurable_space" title="Measurable space">Measurable space</a>/<a href="/wiki/Measurable_function" title="Measurable function">function</a></li> <li><a href="/wiki/Minkowski_distance" title="Minkowski distance">Minkowski distance</a></li> <li><a href="/wiki/Sequence_space" title="Sequence space">Sequence spaces</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L1_space" class="mw-redirect" title="L1 space"><i>L</i><sup>1</sup> spaces</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/Integrable_function" class="mw-redirect" title="Integrable function">Integrable function</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li> <li><a href="/wiki/Taxicab_geometry" title="Taxicab geometry">Taxicab geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L2_space" class="mw-redirect" title="L2 space"><i>L</i><sup>2</sup> spaces</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/Bessel%27s_inequality" title="Bessel's inequality">Bessel's</a></li> <li><a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz</a></li> <li><a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a></li> <li><a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a></li> <li><a href="/wiki/Polarization_identity" title="Polarization identity">Polarization identity</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li> <li><a href="/wiki/Square-integrable_function" title="Square-integrable function">Square-integrable function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L-infinity" title="L-infinity"><span class="mwe-math-element"><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^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}"></span> spaces</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/Bounded_function" title="Bounded function">Bounded function</a></li> <li><a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a></li> <li><a href="/wiki/Infimum_and_supremum" title="Infimum and supremum">Infimum and supremum</a> <ul><li><a href="/wiki/Essential_infimum_and_essential_supremum" title="Essential infimum and essential supremum">Essential</a></li></ul></li> <li><a class="mw-selflink selflink">Uniform norm</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/Almost_everywhere" title="Almost everywhere">Almost everywhere</a></li> <li><a href="/wiki/Convergence_almost_everywhere" class="mw-redirect" title="Convergence almost everywhere">Convergence almost everywhere</a></li> <li><a href="/wiki/Convergence_in_measure" title="Convergence in measure">Convergence in measure</a></li> <li><a href="/wiki/Function_space" title="Function space">Function space</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a></li> <li><a href="/wiki/Locally_integrable_function" title="Locally integrable function">Locally integrable function</a></li> <li><a href="/wiki/Measurable_function" title="Measurable function">Measurable function</a></li> <li><a href="/wiki/Symmetric_decreasing_rearrangement" title="Symmetric decreasing rearrangement">Symmetric decreasing rearrangement</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Inequalities</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/Babenko%E2%80%93Beckner_inequality" title="Babenko–Beckner inequality">Babenko–Beckner</a></li> <li><a href="/wiki/Chebyshev%27s_inequality" title="Chebyshev's inequality">Chebyshev's</a></li> <li><a href="/wiki/Clarkson%27s_inequalities" title="Clarkson's inequalities">Clarkson's</a></li> <li><a href="/wiki/Hanner%27s_inequalities" title="Hanner's inequalities">Hanner's</a></li> <li><a href="/wiki/Hausdorff%E2%80%93Young_inequality" title="Hausdorff–Young inequality">Hausdorff–Young</a></li> <li><a href="/wiki/H%C3%B6lder%27s_inequality" title="Hölder's inequality">Hölder's</a></li> <li><a href="/wiki/Markov%27s_inequality" title="Markov's inequality">Markov's</a></li> <li><a href="/wiki/Minkowski_inequality" title="Minkowski inequality">Minkowski</a></li> <li><a href="/wiki/Young%27s_convolution_inequality" title="Young's convolution inequality">Young's convolution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_analysis" title="Category:Theorems in analysis">Results</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/Marcinkiewicz_interpolation_theorem" title="Marcinkiewicz interpolation theorem">Marcinkiewicz interpolation theorem</a></li> <li><a href="/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a></li> <li><a href="/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue</a></li> <li><a href="/wiki/Riesz%E2%80%93Fischer_theorem" title="Riesz–Fischer theorem">Riesz–Fischer theorem</a></li> <li><a href="/wiki/Riesz%E2%80%93Thorin_theorem" title="Riesz–Thorin theorem">Riesz–Thorin theorem</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><span style="font-size:85%;">For <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a></span></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/Isoperimetric_inequality" title="Isoperimetric inequality">Isoperimetric inequality</a></li> <li><a href="/wiki/Brunn%E2%80%93Minkowski_theorem" title="Brunn–Minkowski theorem">Brunn–Minkowski theorem</a> <ul><li><a href="/wiki/Milman%27s_reverse_Brunn%E2%80%93Minkowski_inequality" title="Milman's reverse Brunn–Minkowski inequality">Milman's reverse</a></li></ul></li> <li><a href="/wiki/Minkowski%E2%80%93Steiner_formula" title="Minkowski–Steiner formula">Minkowski–Steiner formula</a></li> <li><a href="/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality" title="Prékopa–Leindler inequality">Prékopa–Leindler inequality</a></li> <li><a href="/wiki/Vitale%27s_random_Brunn%E2%80%93Minkowski_inequality" title="Vitale's random Brunn–Minkowski inequality">Vitale's random Brunn–Minkowski inequality</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications & related</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/Bochner_space" title="Bochner space">Bochner space</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Lorentz_space" title="Lorentz space">Lorentz space</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></li> <li><a href="/wiki/Quasinorm" title="Quasinorm">Quasinorm</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev space</a></li> <li><a href="/wiki/*-algebra" title="*-algebra">*-algebra</a> <ul><li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann</a></li></ul></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="Banach_space_topics" style="padding:3px"><table class="nowraplinks 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:Banach_spaces" title="Template:Banach spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Banach_spaces" title="Template talk:Banach spaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Banach_spaces" title="Special:EditPage/Template:Banach spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Banach_space_topics" style="font-size:114%;margin:0 4em"><a href="/wiki/Banach_space" title="Banach space">Banach space</a> topics</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of Banach spaces</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Asplund_space" title="Asplund space">Asplund</a></li> <li><a href="/wiki/Banach_space" title="Banach space">Banach</a> <ul><li><a href="/wiki/List_of_Banach_spaces" title="List of Banach spaces">list</a></li></ul></li> <li><a href="/wiki/Banach_lattice" title="Banach lattice">Banach lattice</a></li> <li><a href="/wiki/Grothendieck_space" title="Grothendieck space">Grothendieck </a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a> <ul><li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></li> <li><a href="/wiki/Polarization_identity" title="Polarization identity">Polarization identity</a></li></ul></li> <li>(<a href="/wiki/Polynomially_reflexive_space" title="Polynomially reflexive space">Polynomially</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/L-semi-inner_product" title="L-semi-inner product">L-semi-inner product</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/Uniformly_smooth_space" title="Uniformly smooth space">Uniformly smooth</a></li> <li>(<a href="/wiki/Injective_tensor_product" title="Injective tensor product">Injective</a></li> <li><a href="/wiki/Projective_tensor_product" title="Projective tensor product">Projective</a>) <a href="/wiki/Topological_tensor_product" title="Topological tensor product">Tensor product</a> (<a href="/wiki/Tensor_product_of_Hilbert_spaces" title="Tensor product of Hilbert spaces">of Hilbert spaces</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Banach spaces are:</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/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/F-space" title="F-space">F-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</a></li></ul></li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a> <ul><li><a href="/wiki/Locally_convex_topological_vector_space#Definition_via_seminorms" title="Locally convex topological vector space">Seminorms</a>/<a href="/wiki/Minkowski_functional" title="Minkowski functional">Minkowski functionals</a></li></ul></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">Metrizable</a></li> <li><a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">Normed</a> <ul><li><a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a></li></ul></li> <li><a href="/wiki/Quasinorm" title="Quasinorm">Quasinormed</a></li> <li><a href="/wiki/Stereotype_space" class="mw-redirect" title="Stereotype space">Stereotype</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Function space Topologies</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach%E2%80%93Mazur_compactum" title="Banach–Mazur compactum">Banach–Mazur compactum</a></li> <li><a href="/wiki/Dual_topology" title="Dual topology">Dual</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual space</a> <ul><li><a href="/wiki/Dual_norm" title="Dual norm">Dual norm</a></li></ul></li> <li><a href="/wiki/Operator_topologies" title="Operator topologies">Operator</a></li> <li><a href="/wiki/Ultraweak_topology" title="Ultraweak topology">Ultraweak</a></li> <li><a href="/wiki/Weak_topology" title="Weak topology">Weak</a> <ul><li><a href="/wiki/Weak_topology_(polar_topology)" class="mw-redirect" title="Weak topology (polar topology)">polar</a></li> <li><a href="/wiki/Weak_operator_topology" title="Weak operator topology">operator</a></li></ul></li> <li><a href="/wiki/Strong_topology" title="Strong topology">Strong</a> <ul><li><a href="/wiki/Strong_topology_(polar_topology)" class="mw-redirect" title="Strong topology (polar topology)">polar</a></li> <li><a href="/wiki/Strong_operator_topology" title="Strong operator topology">operator</a></li></ul></li> <li><a href="/wiki/Ultrastrong_topology" title="Ultrastrong topology">Ultrastrong</a></li> <li><a href="/wiki/Topology_of_uniform_convergence" class="mw-redirect" title="Topology of uniform convergence">Uniform convergence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">Linear operators</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">Adjoint</a></li> <li><a href="/wiki/Bilinear_map" title="Bilinear map">Bilinear</a> <ul><li><a href="/wiki/Bilinear_form" title="Bilinear form">form</a></li> <li><a href="/wiki/Bilinear_map" title="Bilinear map">operator</a></li> <li><a href="/wiki/Sesquilinear_form" title="Sesquilinear form">sesquilinear</a></li></ul></li> <li>(<a href="/wiki/Unbounded_operator" title="Unbounded operator">Un</a>)<a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</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> <ul><li><a href="/wiki/Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">on Hilbert spaces</a></li></ul></li> <li>(<a href="/wiki/Discontinuous_linear_map" title="Discontinuous linear map">Dis</a>)<a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">Continuous</a></li> <li><a href="/wiki/Densely_defined" class="mw-redirect" title="Densely defined">Densely defined</a></li> <li>Fredholm <ul><li><a href="/wiki/Fredholm_kernel" title="Fredholm kernel">kernel</a></li> <li><a href="/wiki/Fredholm_operator" title="Fredholm operator">operator</a></li></ul></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Linear_form" title="Linear form">Functionals</a> <ul><li><a href="/wiki/Positive_linear_functional" title="Positive linear functional">positive</a></li></ul></li> <li><a href="/wiki/Pseudo-monotone_operator" title="Pseudo-monotone operator">Pseudo-monotone</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/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint</a></li> <li><a href="/wiki/Strictly_singular_operator" title="Strictly singular operator">Strictly singular</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/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operator_theory" title="Operator theory">Operator theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebras</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebras</a></li> <li><a href="/wiki/Operator_space" title="Operator space">Operator space</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">Spectrum</a> <ul><li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Spectral_radius" title="Spectral radius">radius</a></li></ul></li> <li><a href="/wiki/Spectral_theory" title="Spectral theory">Spectral theory</a> <ul><li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">of ODEs</a></li> <li><a href="/wiki/Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li></ul></li> <li><a href="/wiki/Polar_decomposition" title="Polar decomposition">Polar decomposition</a></li> <li><a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><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-even hlist" 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/Banach%E2%80%93Mazur_theorem" title="Banach–Mazur theorem">Banach–Mazur</a></li> <li><a href="/wiki/Banach%E2%80%93Saks_theorem" class="mw-redirect" title="Banach–Saks theorem">Banach–Saks</a></li> <li><a href="/wiki/Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">Banach–Schauder (open mapping)</a></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Banach–Steinhaus (Uniform boundedness)</a></li> <li><a href="/wiki/Bessel%27s_inequality" title="Bessel's inequality">Bessel's inequality</a></li> <li><a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a></li> <li><a href="/wiki/Closed_graph_theorem" title="Closed graph theorem">Closed graph</a></li> <li><a href="/wiki/Closed_range_theorem" title="Closed range theorem">Closed range</a></li> <li><a href="/wiki/Eberlein%E2%80%93%C5%A0mulian_theorem" title="Eberlein–Šmulian theorem">Eberlein–Šmulian</a></li> <li><a href="/wiki/Freudenthal_spectral_theorem" title="Freudenthal spectral theorem">Freudenthal spectral</a></li> <li><a href="/wiki/Gelfand%E2%80%93Mazur_theorem" title="Gelfand–Mazur theorem">Gelfand–Mazur</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li> <li><a href="/wiki/Goldstine_theorem" title="Goldstine theorem">Goldstine</a></li> <li><a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a> <ul><li><a href="/wiki/Hyperplane_separation_theorem" title="Hyperplane separation theorem">hyperplane separation</a></li></ul></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/Invariant_subspace_problem#Known_special_cases" title="Invariant subspace problem">Lomonosov's invariant subspace</a></li> <li><a href="/wiki/Mackey%E2%80%93Arens_theorem" title="Mackey–Arens theorem">Mackey–Arens</a></li> <li><a href="/wiki/Mazur%27s_lemma" title="Mazur's lemma">Mazur's lemma</a></li> <li><a href="/wiki/M._Riesz_extension_theorem" title="M. Riesz extension theorem">M. Riesz extension</a></li> <li><a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a></li> <li><a href="/wiki/Riesz%27s_lemma" title="Riesz's lemma">Riesz's lemma</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li> <li><a href="/wiki/Ursescu_theorem#Robinson–Ursescu_theorem" title="Ursescu theorem">Robinson-Ursescu</a></li> <li><a href="/wiki/Schauder_fixed-point_theorem" title="Schauder fixed-point theorem">Schauder fixed-point</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Analysis</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract Wiener space</a></li> <li><a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifold</a> <ul><li><a href="/wiki/Banach_bundle" title="Banach bundle">bundle</a></li></ul></li> <li><a href="/wiki/Bochner_space" title="Bochner space">Bochner space</a></li> <li><a href="/wiki/Convex_series" title="Convex series">Convex series</a></li> <li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces" title="Differentiation in Fréchet spaces">Differentiation in Fréchet spaces</a></li> <li><a href="/wiki/Derivative" title="Derivative">Derivatives</a> <ul><li><a href="/wiki/Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet</a></li> <li><a href="/wiki/Gateaux_derivative" title="Gateaux derivative">Gateaux</a></li> <li><a href="/wiki/Functional_derivative" title="Functional derivative">functional</a></li> <li><a href="/wiki/Infinite-dimensional_holomorphy" title="Infinite-dimensional holomorphy">holomorphic</a></li> <li><a href="/wiki/Quasi-derivative" title="Quasi-derivative">quasi</a></li></ul></li> <li><a href="/wiki/Integral" title="Integral">Integrals</a> <ul><li><a href="/wiki/Bochner_integral" title="Bochner integral">Bochner</a></li> <li><a href="/wiki/Dunford_integral" class="mw-redirect" title="Dunford integral">Dunford</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">Gelfand–Pettis</a></li> <li><a href="/wiki/Regulated_integral" title="Regulated integral">regulated</a></li> <li><a href="/wiki/Paley%E2%80%93Wiener_integral" title="Paley–Wiener integral">Paley–Wiener</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">weak</a></li></ul></li> <li><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a> <ul><li><a href="/wiki/Borel_functional_calculus" title="Borel functional calculus">Borel</a></li> <li><a href="/wiki/Continuous_functional_calculus" title="Continuous functional calculus">continuous</a></li> <li><a href="/wiki/Holomorphic_functional_calculus" title="Holomorphic functional calculus">holomorphic</a></li></ul></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measures</a> <ul><li><a href="/wiki/Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">Lebesgue</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li> <li><a href="/wiki/Vector_measure" title="Vector measure">Vector</a></li></ul></li> <li><a href="/wiki/Weakly_measurable_function" title="Weakly measurable function">Weakly</a> / <a href="/wiki/Strongly_measurable_functions" class="mw-redirect" title="Strongly measurable functions">Strongly</a> measurable function</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of sets</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">Absolutely convex</a></li> <li><a href="/wiki/Absorbing_set" title="Absorbing set">Absorbing</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/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Bounded</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/Convex_series#Types_of_subsets" title="Convex series">Convex series related</a> ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (H<i>x</i>), and (Hw<i>x</i>))</li> <li><a href="/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/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> <li><a href="/wiki/Zonotope" class="mw-redirect" title="Zonotope">Zonotope</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Subsets / set operations</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/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/Bounding_point" title="Bounding point">Bounding points</a></li> <li><a href="/wiki/Convex_hull" title="Convex hull">Convex hull</a></li> <li><a href="/wiki/Extreme_point" title="Extreme point">Extreme point</a></li> <li><a href="/wiki/Interior_(topology)" title="Interior (topology)">Interior</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%"><a href="/wiki/Template:ListOfBanachSpaces" class="mw-redirect" title="Template:ListOfBanachSpaces">Examples</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_continuity" title="Absolute continuity">Absolute continuity <i>AC</i></a></li> <li><a href="/wiki/Ba_space" title="Ba 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 ba(\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mi>a</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ba(\Sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58fe61351e3531b14043fa2d09e98c2437bd1a6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.715ex; height:2.843ex;" alt="{\displaystyle ba(\Sigma )}"></span></a></li> <li><a href="/wiki/C_space" title="C space">c space</a></li> <li><a href="/wiki/BK-space" title="BK-space">Banach coordinate <i>BK</i></a></li> <li><a href="/wiki/Besov_space" title="Besov space">Besov <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p,q}^{s}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{p,q}^{s}(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9919cf78ad095c237169772d2b27a37bfbef1b75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.524ex; height:3.009ex;" alt="{\displaystyle B_{p,q}^{s}(\mathbb {R} )}"></span></a></li> <li><a href="/wiki/Birnbaum%E2%80%93Orlicz_space" class="mw-redirect" title="Birnbaum–Orlicz space">Birnbaum–Orlicz</a></li> <li><a href="/wiki/Bounded_variation" title="Bounded variation">Bounded variation <i>BV</i></a></li> <li><a href="/wiki/Bs_space" title="Bs space">Bs space</a></li> <li><a href="/wiki/Continuous_functions_on_a_compact_Hausdorff_space" title="Continuous functions on a compact Hausdorff space">Continuous <i>C(K)</i> with <i>K</i> compact Hausdorff</a></li> <li><a href="/wiki/Hardy_space" title="Hardy space">Hardy H<sup><i>p</i></sup></a></li> <li><a href="/wiki/Hilbert_space#Definition" title="Hilbert space">Hilbert <i>H</i></a></li> <li><a href="/wiki/Morrey%E2%80%93Campanato_space" title="Morrey–Campanato space">Morrey–Campanato <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\lambda ,p}(\Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\lambda ,p}(\Omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8af58fa038369c3ec6386c6656aab82825e372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.545ex; height:3.176ex;" alt="{\displaystyle L^{\lambda ,p}(\Omega )}"></span></a></li> <li><a href="/wiki/Sequence_space#ℓp_spaces" title="Sequence space"><i>ℓ<sup>p</sup></i></a> <ul><li><a href="/wiki/L-infinity#Sequence_space" title="L-infinity"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8348195cf09473662c6f59e6717722a6fc01d0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}"></span></a></li></ul></li> <li><a href="/wiki/Lp_space" title="Lp space"><i>L<sup>p</sup></i></a> <ul><li><a href="/wiki/L-infinity#Function_space" title="L-infinity"><span class="mwe-math-element"><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^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}"></span></a></li> <li><a href="/wiki/Lp_space#Weighted_Lp_spaces" title="Lp space">weighted</a></li></ul></li> <li><a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\left(\mathbb {R} ^{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\left(\mathbb {R} ^{n}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0465acd58a0f31e32b095aed742d9ccc6331369c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.592ex; height:2.843ex;" alt="{\displaystyle S\left(\mathbb {R} ^{n}\right)}"></span></a></li> <li><a href="/wiki/Segal%E2%80%93Bargmann_space" title="Segal–Bargmann space">Segal–Bargmann <i>F</i></a></li> <li><a href="/wiki/Sequence_space" title="Sequence space">Sequence space</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev W<sup><i>k,p</i></sup></a> <ul><li><a href="/wiki/Sobolev_inequality" title="Sobolev inequality">Sobolev inequality</a></li></ul></li> <li><a href="/wiki/Triebel%E2%80%93Lizorkin_space" title="Triebel–Lizorkin space">Triebel–Lizorkin</a></li> <li><a href="/wiki/Wiener_amalgam_space" title="Wiener amalgam space">Wiener amalgam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(X,L^{p})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(X,L^{p})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b37b1dc9714960c525cb561a4828f41feb5844ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.9ex; height:2.843ex;" alt="{\displaystyle W(X,L^{p})}"></span></a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element method</a></li> <li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Mathematical formulation of quantum mechanics</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Ordinary Differential Equations (ODEs)</a></li> <li><a href="/wiki/Validated_numerics" title="Validated numerics">Validated numerics</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="Functional_analysis_(topics_–_glossary)" 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: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)" 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 href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">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_operator" title="Integral operator">Integral 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><noscript><img 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