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Supremumnormen – Wikipedia
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class="vector-toc-link" href="#Exempel"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Exempel</span> </div> </a> <ul id="toc-Exempel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Väsentlig_supremumnorm" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Väsentlig_supremumnorm"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Väsentlig supremumnorm</span> </div> </a> <button aria-controls="toc-Väsentlig_supremumnorm-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>Växla underavsnittet Väsentlig supremumnorm</span> </button> <ul id="toc-Väsentlig_supremumnorm-sublist" class="vector-toc-list"> <li id="toc-Normerade_och_seminormerade_rum_med_väsentliga_supremumnormen" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Normerade_och_seminormerade_rum_med_väsentliga_supremumnormen"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Normerade och seminormerade rum med väsentliga supremumnormen</span> </div> </a> <ul id="toc-Normerade_och_seminormerade_rum_med_väsentliga_supremumnormen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_till_andra_normer" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_till_andra_normer"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Relation till andra normer</span> </div> </a> <ul id="toc-Relation_till_andra_normer-sublist" class="vector-toc-list"> <li id="toc-Bevis" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Bevis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Bevis</span> </div> </a> <ul id="toc-Bevis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Innehåll" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Växla innehållsförteckningen" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Växla innehållsförteckningen</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Supremumnormen</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Gå till en artikel på ett annat språk. Tillgänglig på 9 språk" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-9" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">9 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%83%D0%BF%D1%80%D0%B5%D0%BC%D1%83%D0%BC-%D0%BD%D0%BE%D1%80%D0%BC%D0%B0" title="Супремум-норма – bulgariska" lang="bg" hreflang="bg" data-title="Супремум-норма" data-language-autonym="Български" data-language-local-name="bulgariska" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Supremumsnorm" title="Supremumsnorm – tyska" lang="de" hreflang="de" data-title="Supremumsnorm" data-language-autonym="Deutsch" data-language-local-name="tyska" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Uniform_norm" title="Uniform norm – engelska" lang="en" hreflang="en" data-title="Uniform norm" data-language-autonym="English" data-language-local-name="engelska" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Norma_del_supremo" title="Norma del supremo – spanska" lang="es" hreflang="es" data-title="Norma del supremo" data-language-autonym="Español" data-language-local-name="spanska" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-ko badge-Q70893996 mw-list-item" title=""><a href="https://ko.wikipedia.org/wiki/%EA%B7%A0%EB%93%B1_%EB%85%B8%EB%A6%84" title="균등 노름 – koreanska" lang="ko" hreflang="ko" data-title="균등 노름" data-language-autonym="한국어" data-language-local-name="koreanska" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Norma_uniforme" title="Norma uniforme – italienska" lang="it" hreflang="it" data-title="Norma uniforme" data-language-autonym="Italiano" data-language-local-name="italienska" 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 – japanska" lang="ja" hreflang="ja" data-title="一様ノルム" data-language-autonym="日本語" data-language-local-name="japanska" class="interlanguage-link-target"><span>日本語</span></a></li><li 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href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fsv.wikipedia.org%2Fwiki%2FSupremumnormen"><span>Hämta förkortad url</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fsv.wikipedia.org%2Fwiki%2FSupremumnormen"><span>Ladda ner QR-kod</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Skriv ut/exportera </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Special:Bok&bookcmd=book_creator&referer=Supremumnormen"><span>Skapa en bok</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Supremumnormen&action=show-download-screen"><span>Ladda ned som PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Supremumnormen&printable=yes" title="Utskriftsvänlig version av den här sidan [p]" accesskey="p"><span>Utskriftsvänlig version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> På andra projekt </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1202673" title="Länk till anslutet databasobjekt [g]" accesskey="g"><span>Wikidata-objekt</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Sidverktyg"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Utseende"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Utseende</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">flytta till sidofältet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">dölj</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Från Wikipedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="sv" dir="ltr"><p><b>Supremumnormen</b>, även kallad <b>Tjebysjovnormen</b> eller informellt <b>oändlighetsnormen</b>, är inom <a href="/wiki/Matematisk_analys" title="Matematisk analys">matematisk analys</a> en <a href="/wiki/Norm_(matematik)" title="Norm (matematik)">norm</a> för <a href="/wiki/Funktion" title="Funktion">funktioner</a>. Normen tilldelar ett <a href="/wiki/Reella_tal" title="Reella tal">reellt</a> positivt tal till en reell eller <a href="/wiki/Komplex_funktion" class="mw-redirect" title="Komplex funktion">komplex funktion</a>. Förenklat kan man säga att supremumnormen mäter "storleken" på en funktion. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_och_användning"><span id="Definition_och_anv.C3.A4ndning"></span>Definition och användning</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supremumnormen&veaction=edit&section=1" title="Redigera avsnitt: Definition och användning" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Supremumnormen&action=edit&section=1" title="Redigera avsnitts källkod: Definition och användning"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Låt <i>X</i> vara en mängd och <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{X}:=\{f|f:X\rightarrow \mathbb {R} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→<!-- → --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{X}:=\{f|f:X\rightarrow \mathbb {R} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cefce21799f639e229c53604b80d0f930150b0b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.794ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} ^{X}:=\{f|f:X\rightarrow \mathbb {R} \}}"></span>. <b>Supremumnormen</b> för <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in \mathbb {R} ^{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈<!-- ∈ --></mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in \mathbb {R} ^{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63c2419f0d53c2ad533ee08e922b452dea18b0a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.43ex; height:3.009ex;" alt="{\displaystyle f\in \mathbb {R} ^{X}}"></span> är talet </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 }:=\sup\{|f(x)|:x\in X\}}"> <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 movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mi>x</mi> <mo>∈<!-- ∈ --></mo> <mi>X</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }:=\sup\{|f(x)|:x\in X\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3005301eb99ba0d292f54b2f54121e57e043d86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.849ex; height:2.843ex;" alt="{\displaystyle \|f\|_{\infty }:=\sup\{|f(x)|:x\in X\}}"></span>.</dd></dl> <p>Fast <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mo>⋅<!-- ⋅ --></mo> <msub> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b723232adf7317abb1fc1c1326e1e4f79616a7e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.879ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{\infty }}"></span> kallas supremum<i>normen</i> är detta inte alltid en <a href="/wiki/Norm_(matematik)" title="Norm (matematik)">norm</a> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{X}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{X}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b5a1e4cb747acdcbdd9c26e9f745851d9dece2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.698ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{X}\,}"></span>. T. ex. om <span class="mwe-math-element"><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=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2e2b6427cd2b517be352b378a1830c1540e3a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.757ex; height:2.176ex;" alt="{\displaystyle X=\mathbb {R} }"></span> vi har </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\mapsto x\|_{\infty }=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>x</mi> <mo 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> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\mapsto x\|_{\infty }=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b29ee434037d6e15c21c0cce34ac6a6a043bfe31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.896ex; height:2.843ex;" alt="{\displaystyle \|x\mapsto x\|_{\infty }=\infty }"></span></dd></dl> <p>men normen måste vara ändlig. Så man får istället definiera mängden av alla <a href="/wiki/Begr%C3%A4nsad_funktion" title="Begränsad funktion">begränsade funktioner</a>: </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 {\mathcal {B}}(X,\mathbb {R} ):=\{f\in \mathbb {R} ^{X}:\|f\|_{\infty }<\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>∈<!-- ∈ --></mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <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> <mi mathvariant="normal">∞<!-- ∞ --></mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}(X,\mathbb {R} ):=\{f\in \mathbb {R} ^{X}:\|f\|_{\infty }<\infty \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ad2ee68a833a4424068d836149cc9b1c7a17306" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.383ex; height:3.176ex;" alt="{\displaystyle {\mathcal {B}}(X,\mathbb {R} ):=\{f\in \mathbb {R} ^{X}:\|f\|_{\infty }<\infty \}}"></span></dd></dl> <p>då supremumnormen är en norm, dvs paret <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {B}}(X,\mathbb {R} ),\|\cdot \|_{\infty })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mo>⋅<!-- ⋅ --></mo> <msub> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {B}}(X,\mathbb {R} ),\|\cdot \|_{\infty })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86254e4d5cd0ae6ba79360a3a7bb07b8c3f73b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.767ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {B}}(X,\mathbb {R} ),\|\cdot \|_{\infty })}"></span> är ett <a href="/wiki/Norm_(matematik)" title="Norm (matematik)">normerat rum</a>. Det här är ett resultat från <a href="/wiki/Absolutbelopp" title="Absolutbelopp">absolutbeloppets</a> egenskaper. </p><p>Man kan inducera en <a href="/wiki/Metrik_(matematik)" class="mw-redirect" title="Metrik (matematik)">metrik</a> från supremumnormen som mäter avståndet mellan två begränsade funktioner: </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):=\|f-g\|_{\infty }}"> <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> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>f</mi> <mo>−<!-- − --></mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(f,g):=\|f-g\|_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ffbbc038048a2d58223430f55e45060ad821980" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.634ex; height:2.843ex;" alt="{\displaystyle d(f,g):=\|f-g\|_{\infty }}"></span>.</dd></dl> <p>Så att en <a href="/wiki/F%C3%B6ljd" title="Följd">följd</a> av funktioner, <span class="mwe-math-element"><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>, <a href="/wiki/Likformig_konvergens" title="Likformig konvergens">konvergerar likformigt</a> till en funktion <span class="mwe-math-element"><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> <a href="/wiki/Om_och_endast_om" title="Om och endast om">om och endast om</a> </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 \lim _{n\rightarrow \infty }\|f_{n}-f\|_{\infty }=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> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−<!-- − --></mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty }\|f_{n}-f\|_{\infty }=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f56a84f3e1fad214123ea07f1a8bd75e77536a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.244ex; height:3.843ex;" alt="{\displaystyle \lim _{n\rightarrow \infty }\|f_{n}-f\|_{\infty }=0.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Exempel">Exempel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supremumnormen&veaction=edit&section=2" title="Redigera avsnitt: Exempel" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Supremumnormen&action=edit&section=2" title="Redigera avsnitts källkod: Exempel"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fil:Vector_norm_sup.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Vector_norm_sup.svg/250px-Vector_norm_sup.svg.png" decoding="async" width="250" height="252" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Vector_norm_sup.svg/375px-Vector_norm_sup.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Vector_norm_sup.svg/500px-Vector_norm_sup.svg.png 2x" data-file-width="144" data-file-height="145" /></a><figcaption>Element <i>x</i> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> med <span class="mwe-math-element"><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 }=k}"> <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> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{\infty }=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a64ec4d987a393c8978120628acdcb4bc4916a7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.84ex; height:2.843ex;" alt="{\displaystyle \|x\|_{\infty }=k}"></span>, där <i>k</i> är en konstant.</figcaption></figure> <ul><li>Om <i>X</i> är ett <a href="/wiki/Kompakt" class="mw-redirect" title="Kompakt">kompakt</a> <a href="/wiki/Topologiskt_rum" class="mw-redirect" title="Topologiskt rum">topologiskt rum</a>, exempelvis <a href="/wiki/Intervall_(matematik)" title="Intervall (matematik)">intervallet</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 [0,1]\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e2b417e116123c724ee6f69cf309f6ad17a2d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.04ex; height:2.843ex;" alt="{\displaystyle [0,1]\,}"></span>, är mängden av alla <a href="/wiki/Kontinuerlig_funktion" title="Kontinuerlig funktion">kontinuerliga funktioner</a> med supremumnormen, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {C}}(X),\|\cdot \|_{\infty })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mo>⋅<!-- ⋅ --></mo> <msub> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {C}}(X),\|\cdot \|_{\infty })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c85da0f41f0722f14624148d829bf748e88577e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.751ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {C}}(X),\|\cdot \|_{\infty })}"></span>, ett normerat rum.</li></ul> <ul><li>Om <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\{1,2,...,n\}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\{1,2,...,n\}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4afc64c5a7546a58eaaf102038559acc85e3613a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.714ex; height:2.843ex;" alt="{\displaystyle X=\{1,2,...,n\}\,}"></span>, för <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span>, är <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{X}=\mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{X}=\mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4673a6dec08810eda32fa68c26ddadaea448009f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.305ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{X}=\mathbb {R} ^{n}}"></span>. Supremum kan alltså här ersättas med maximum: <span class="mwe-math-element"><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\{|x_{i}|:i\in \{1,2,...,n\}\}}"> <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> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mi>i</mi> <mo>∈<!-- ∈ --></mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{\infty }=\max\{|x_{i}|:i\in \{1,2,...,n\}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc44e580199c92eb1e5dcc3971ced7395914776a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.531ex; height:2.843ex;" alt="{\displaystyle \|x\|_{\infty }=\max\{|x_{i}|:i\in \{1,2,...,n\}\}}"></span> för <span class="mwe-math-element"><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=(x_{1},x_{2},...,x_{n})\in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <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> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=(x_{1},x_{2},...,x_{n})\in \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b97f5950e74a1caa9be6b99cd70bbeef3e09695" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.494ex; height:2.843ex;" alt="{\displaystyle x=(x_{1},x_{2},...,x_{n})\in \mathbb {R} ^{n}}"></span> och <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ^{n},\|\cdot \|_{\infty })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</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> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mo>⋅<!-- ⋅ --></mo> <msub> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ^{n},\|\cdot \|_{\infty })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3079a2b519982ecc276ca18401dabde5b266e416" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.619ex; height:2.843ex;" alt="{\displaystyle (\mathbb {R} ^{n},\|\cdot \|_{\infty })}"></span> är ett normerat rum.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Väsentlig_supremumnorm"><span id="V.C3.A4sentlig_supremumnorm"></span>Väsentlig supremumnorm</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supremumnormen&veaction=edit&section=3" title="Redigera avsnitt: Väsentlig supremumnorm" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Supremumnormen&action=edit&section=3" title="Redigera avsnitts källkod: Väsentlig supremumnorm"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Om vi har ett <a href="/wiki/M%C3%A5tt_(matematik)" title="Mått (matematik)">måttstruktur</a> i <i>X</i> kan vi generalisera supremumnormen. Låt <span class="mwe-math-element"><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,{\mathcal {F}},\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mi>μ<!-- μ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {F}},\mu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1efec76d53b2977e284b7cf525c5ceffe299e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.185ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {F}},\mu )}"></span> vara ett <a href="/wiki/M%C3%A5tt_(matematik)" title="Mått (matematik)">måttrum</a> och </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 {\mathcal {M}}(X,\mathbb {R} ):=\{f\in \mathbb {R} ^{X}:f{\mbox{ är }}{\mathcal {F}}{\mbox{-mätbara}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>∈<!-- ∈ --></mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <mo>:</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> är </mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>-mätbara</mtext> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}(X,\mathbb {R} ):=\{f\in \mathbb {R} ^{X}:f{\mbox{ är }}{\mathcal {F}}{\mbox{-mätbara}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68879b133d3950645e2a1bb546d93960870e1809" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:40.985ex; height:3.509ex;" alt="{\displaystyle {\mathcal {M}}(X,\mathbb {R} ):=\{f\in \mathbb {R} ^{X}:f{\mbox{ är }}{\mathcal {F}}{\mbox{-mätbara}}\}}"></span>.</dd></dl> <p>Då är <b>väsentliga supremumnormen</b> för <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\mathcal {M}}(X,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</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 f\in {\mathcal {M}}(X,\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11547f90a400d44e0226f89f71452262a30df5ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.411ex; height:2.843ex;" alt="{\displaystyle f\in {\mathcal {M}}(X,\mathbb {R} )}"></span> </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 }^{\operatorname {ess} }:=\operatorname {ess} \sup |f|=\inf\{r\in \mathbb {R} :\mu (\{x\in X:|f(x)|>r\})=0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ess</mi> </mrow> </msubsup> <mo>:=</mo> <mi>ess</mi> <mo>⁡<!-- --></mo> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>r</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <mi>μ<!-- μ --></mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈<!-- ∈ --></mo> <mi>X</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }^{\operatorname {ess} }:=\operatorname {ess} \sup |f|=\inf\{r\in \mathbb {R} :\mu (\{x\in X:|f(x)|>r\})=0\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f8eb005d44c0c53e122fda2ea4aa1cc2f9a2c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.288ex; height:2.843ex;" alt="{\displaystyle \|f\|_{\infty }^{\operatorname {ess} }:=\operatorname {ess} \sup |f|=\inf\{r\in \mathbb {R} :\mu (\{x\in X:|f(x)|>r\})=0\}.}"></span></dd></dl> <p>där <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ess} \sup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ess</mi> <mo>⁡<!-- --></mo> <mo movablelimits="true" form="prefix">sup</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ess} \sup }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b803c1f353eabd24a09517e2c42db9b11ebcdcfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.753ex; height:2.009ex;" alt="{\displaystyle \operatorname {ess} \sup }"></span> är <a href="/wiki/V%C3%A4sentligt_supremum" class="mw-redirect" title="Väsentligt supremum">väsentligt supremum</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Normerade_och_seminormerade_rum_med_väsentliga_supremumnormen"><span id="Normerade_och_seminormerade_rum_med_v.C3.A4sentliga_supremumnormen"></span>Normerade och seminormerade rum med väsentliga supremumnormen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supremumnormen&veaction=edit&section=4" title="Redigera avsnitt: Normerade och seminormerade rum med väsentliga supremumnormen" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Supremumnormen&action=edit&section=4" title="Redigera avsnitts källkod: Normerade och seminormerade rum med väsentliga supremumnormen"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Några egenskaper för väsentliga supremumnormen är: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{\infty }^{\mathrm {ess} }\leq \|f\|_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }^{\mathrm {ess} }\leq \|f\|_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50139a39bf22e35aae66a2ac2d2a19a9e300957c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.439ex; height:2.843ex;" alt="{\displaystyle \|f\|_{\infty }^{\mathrm {ess} }\leq \|f\|_{\infty }}"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|af\|_{\infty }^{\mathrm {ess} }=|a|\|f\|_{\infty }^{\mathrm {ess} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>a</mi> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|af\|_{\infty }^{\mathrm {ess} }=|a|\|f\|_{\infty }^{\mathrm {ess} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f1fe6f0dde5897f1801f23665387da7f4ea640e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.575ex; height:2.843ex;" alt="{\displaystyle \|af\|_{\infty }^{\mathrm {ess} }=|a|\|f\|_{\infty }^{\mathrm {ess} }}"></span> och</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f+g\|_{\infty }^{\mathrm {ess} }\leq \|f\|_{\infty }^{\mathrm {ess} }+\|g\|_{\infty }^{\mathrm {ess} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> <mo>≤<!-- ≤ --></mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> <mo>+</mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>g</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f+g\|_{\infty }^{\mathrm {ess} }\leq \|f\|_{\infty }^{\mathrm {ess} }+\|g\|_{\infty }^{\mathrm {ess} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec0292d7d7ae48525e22f24f95b8c4c112d9514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.317ex; height:2.843ex;" alt="{\displaystyle \|f+g\|_{\infty }^{\mathrm {ess} }\leq \|f\|_{\infty }^{\mathrm {ess} }+\|g\|_{\infty }^{\mathrm {ess} }}"></span></li></ul> <p>för alla <span class="mwe-math-element"><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\in {\mathcal {M}}(X,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</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 f,g\in {\mathcal {M}}(X,\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53834492e103cd258dff9e6c688cf5a9efb3ece0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.561ex; height:2.843ex;" alt="{\displaystyle f,g\in {\mathcal {M}}(X,\mathbb {R} )}"></span> och <span class="mwe-math-element"><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\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b044c60e01b54c7116ee355431f37ed846badc53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle a\in \mathbb {R} }"></span>. Detta ger att <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} ),\|\cdot \|_{\infty }^{\mathrm {ess} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>∩<!-- ∩ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mo>⋅<!-- ⋅ --></mo> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} ),\|\cdot \|_{\infty }^{\mathrm {ess} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30636046ba03c16d4fdee42383a0c99bd2a11801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.024ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} ),\|\cdot \|_{\infty }^{\mathrm {ess} })}"></span> är ett (<a href="/wiki/Seminorm" class="mw-redirect" title="Seminorm">seminormerat rum</a>. </p><p>Seminormen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{\infty }^{\mathrm {ess} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mo>⋅<!-- ⋅ --></mo> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{\infty }^{\mathrm {ess} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd682d2f3df3a1a0a9a6b22e3600383a62d5b97c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.262ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{\infty }^{\mathrm {ess} }}"></span> är inte en norm eftersom det finns funktioner som inte är nollfunktionen men som har en väsentligt supremumnorm som är noll, om exempelvis <span class="mwe-math-element"><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,{\mathcal {F}},\mu )=(\mathbb {R} ,\mathrm {Leb} \mathbb {R} ,{\mathcal {L}}_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mi>μ<!-- μ --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {F}},\mu )=(\mathbb {R} ,\mathrm {Leb} \mathbb {R} ,{\mathcal {L}}_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deac02251b5455b2f84dad00c1374309b49039c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.953ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {F}},\mu )=(\mathbb {R} ,\mathrm {Leb} \mathbb {R} ,{\mathcal {L}}_{1})}"></span> får man att </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 \|\chi _{\mathbb {N} }\|_{\infty }^{\mathrm {ess} }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <msub> <mi>χ<!-- χ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\chi _{\mathbb {N} }\|_{\infty }^{\mathrm {ess} }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6126ceb658ec39d0e1eb57259d9e63f26aa3ecbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.718ex; height:2.843ex;" alt="{\displaystyle \|\chi _{\mathbb {N} }\|_{\infty }^{\mathrm {ess} }=0}"></span></dd></dl> <p>där <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ<!-- χ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/338f03e261e525bfb10f49e9867b80fc4c33e8f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.874ex; height:2.009ex;" alt="{\displaystyle \chi _{\mathbb {N} }}"></span> är <a href="/wiki/Indikatorfunktion" class="mw-redirect" title="Indikatorfunktion">indikatorfunktionen</a> för de naturliga talen. Resultatet ovan fås då <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{1}(\mathbb {N} )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{1}(\mathbb {N} )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4abe52d9dc192253ed3104f9db0e718eb8c31ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.406ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}_{1}(\mathbb {N} )=0}"></span> men </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 \chi _{\mathbb {N} }\neq \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ<!-- χ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo>≠<!-- ≠ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{\mathbb {N} }\neq \mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65e2d26fb521b436eafcae84cee05192f5c63967" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.309ex; height:2.676ex;" alt="{\displaystyle \chi _{\mathbb {N} }\neq \mathbf {0} }"></span>.</dd></dl> <p>Men man kan definiera en <a href="/wiki/Ekvivalensrelation" title="Ekvivalensrelation">ekvivalensrelation</a> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>∩<!-- ∩ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</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 {\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/439941dd4117ba345983858d6bc1f176ad1981ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.919ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} )}"></span> genom att </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\sim g\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∼<!-- ∼ --></mo> <mi>g</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\sim g\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb396ac12b653480b30a923a72d34000749ac7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.88ex; height:2.509ex;" alt="{\displaystyle f\sim g\,}"></span> om och endast om <span class="mwe-math-element"><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 }^{\mathrm {ess} }=\|g\|_{\infty }^{\mathrm {ess} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>g</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }^{\mathrm {ess} }=\|g\|_{\infty }^{\mathrm {ess} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/accf91de08f7b1a1cdfcf8ab3063e41b43a6283f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle \|f\|_{\infty }^{\mathrm {ess} }=\|g\|_{\infty }^{\mathrm {ess} }}"></span></dd></dl> <p>och definiera väsentliga supremumnormen för <a href="/wiki/Ekvivalensklass" title="Ekvivalensklass">ekvivalensklasser</a> </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^{\sim }\|_{\infty }^{\mathrm {ess} }:=\|f\|_{\infty }^{\mathrm {ess} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∼<!-- ∼ --></mo> </mrow> </msup> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> <mo>:=</mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f^{\sim }\|_{\infty }^{\mathrm {ess} }:=\|f\|_{\infty }^{\mathrm {ess} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32254db8ca04907b48814648433b0864bb286c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.021ex; height:2.843ex;" alt="{\displaystyle \|f^{\sim }\|_{\infty }^{\mathrm {ess} }:=\|f\|_{\infty }^{\mathrm {ess} }}"></span></dd></dl> <p>där <span class="mwe-math-element"><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^{\sim }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∼<!-- ∼ --></mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{\sim }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92917f41f5fa614c8ef38b52642611a78300612b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.831ex; height:2.509ex;" alt="{\displaystyle f^{\sim }}"></span> är ekvivalensklassen med representant <i>f</i>: </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^{\sim }:=\{g\in {\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} ):f\sim g\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∼<!-- ∼ --></mo> </mrow> </msup> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>∩<!-- ∩ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>:</mo> <mi>f</mi> <mo>∼<!-- ∼ --></mo> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{\sim }:=\{g\in {\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} ):f\sim g\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd4314bd1f0fd947e292128fbc002f641d4c2d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.854ex; height:2.843ex;" alt="{\displaystyle f^{\sim }:=\{g\in {\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} ):f\sim g\}.}"></span></dd></dl> <p>Med denna struktur fås att <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} )/\sim ,\|\cdot \|_{\infty }^{\mathrm {ess} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>∩<!-- ∩ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>∼<!-- ∼ --></mo> <mo>,</mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mo>⋅<!-- ⋅ --></mo> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} )/\sim ,\|\cdot \|_{\infty }^{\mathrm {ess} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7cbf5e22dca470760b41bc9fa1d6716a19ea42e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.64ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} )/\sim ,\|\cdot \|_{\infty }^{\mathrm {ess} })}"></span> är ett normerat rum. </p><p>En fördel med väsentliga supremumnormen är att man kan få med fler funktioner i sitt normerade rum, då det finns måttrum <span class="mwe-math-element"><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,{\mathcal {F}},\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mi>μ<!-- μ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {F}},\mu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1efec76d53b2977e284b7cf525c5ceffe299e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.185ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {F}},\mu )}"></span> och funktioner <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\mathcal {M}}(X,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</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 f\in {\mathcal {M}}(X,\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11547f90a400d44e0226f89f71452262a30df5ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.411ex; height:2.843ex;" alt="{\displaystyle f\in {\mathcal {M}}(X,\mathbb {R} )}"></span> som har <span class="mwe-math-element"><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 }^{\mathrm {ess} }<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> <mo><</mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }^{\mathrm {ess} }<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42eaa37d6a8e1edb54f7533d2ab13e9079f6d8d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.284ex; height:2.843ex;" alt="{\displaystyle \|f\|_{\infty }^{\mathrm {ess} }<\infty }"></span> men <span class="mwe-math-element"><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 }=\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 mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2bfedd946dfa89c744cc0af3f0136888abb859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.901ex; height:2.843ex;" alt="{\displaystyle \|f\|_{\infty }=\infty }"></span>. </p><p>Till exempel, om <span class="mwe-math-element"><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,{\mathcal {F}},\mu )=(\mathbb {R} ,\mathrm {Leb} \mathbb {R} ,{\mathcal {L}}_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mi>μ<!-- μ --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {F}},\mu )=(\mathbb {R} ,\mathrm {Leb} \mathbb {R} ,{\mathcal {L}}_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deac02251b5455b2f84dad00c1374309b49039c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.953ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {F}},\mu )=(\mathbb {R} ,\mathrm {Leb} \mathbb {R} ,{\mathcal {L}}_{1})}"></span> får man att </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 \|\mathbf {1} _{\mathbb {N} }\cdot \exp \|_{\infty }^{\mathrm {ess} }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo>⋅<!-- ⋅ --></mo> <mi>exp</mi> <mo>⁡<!-- --></mo> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {1} _{\mathbb {N} }\cdot \exp \|_{\infty }^{\mathrm {ess} }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e455f9fd513a5d0b4f1a0ea01a952354c60897d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.218ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {1} _{\mathbb {N} }\cdot \exp \|_{\infty }^{\mathrm {ess} }=0}"></span></dd></dl> <p>eftersom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{1}(\mathbb {N} )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{1}(\mathbb {N} )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4abe52d9dc192253ed3104f9db0e718eb8c31ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.406ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}_{1}(\mathbb {N} )=0}"></span> men </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 \|\mathbf {1} _{\mathbb {N} }\cdot \exp \|_{\infty }=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo>⋅<!-- ⋅ --></mo> <mi>exp</mi> <mo>⁡<!-- --></mo> <msub> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {1} _{\mathbb {N} }\cdot \exp \|_{\infty }=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ef70dbe69a2a49a64b43779329279f745230a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.997ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {1} _{\mathbb {N} }\cdot \exp \|_{\infty }=\infty }"></span></dd></dl> <p>eftersom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(n)\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(n)\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6372bba8ff58a269792c2f2b95573d88eee3372c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.694ex; height:2.843ex;" alt="{\displaystyle \exp(n)\rightarrow \infty }"></span> när <span class="mwe-math-element"><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\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9702f04f2d0e5b887b99faeeffb0c4cfd8263eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\rightarrow \infty }"></span>. </p><p>Följaktligen kan man generalisera <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>∩<!-- ∩ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</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 {\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/439941dd4117ba345983858d6bc1f176ad1981ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.919ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} )}"></span>. Låt </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 L^{\infty }=L^{\infty }(X,{\mathcal {F}},\mu ):=\{f\in {\mathcal {M}}(X,\mathbb {R} ):\|f\|_{\infty }^{\mathrm {ess} }<\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> <mo>=</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mi>μ<!-- μ --></mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>:</mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> <mo><</mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }=L^{\infty }(X,{\mathcal {F}},\mu ):=\{f\in {\mathcal {M}}(X,\mathbb {R} ):\|f\|_{\infty }^{\mathrm {ess} }<\infty \}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be9835f852aae6032fe2f3b3e0fc3f14560d0a73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.55ex; height:2.843ex;" alt="{\displaystyle L^{\infty }=L^{\infty }(X,{\mathcal {F}},\mu ):=\{f\in {\mathcal {M}}(X,\mathbb {R} ):\|f\|_{\infty }^{\mathrm {ess} }<\infty \}.}"></span></dd></dl> <p>Så att </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 {\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} )\subset L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>∩<!-- ∩ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>⊂<!-- ⊂ --></mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} )\subset L^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d2acbb2780c3f1aaddca2d87a9950fadaa11f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.476ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}(X,\mathbb {R} )\cap {\mathcal {M}}(X,\mathbb {R} )\subset L^{\infty }}"></span></dd></dl> <p>och <span class="mwe-math-element"><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 },\|\cdot \|_{\infty }^{\mathrm {ess} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msup> <mo>,</mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mo>⋅<!-- ⋅ --></mo> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (L^{\infty },\|\cdot \|_{\infty }^{\mathrm {ess} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1da85f47ea44f56bb213504851b23c3d49484b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.564ex; height:2.843ex;" alt="{\displaystyle (L^{\infty },\|\cdot \|_{\infty }^{\mathrm {ess} })}"></span> är ett seminormerat rum. Man kan transformera <span class="mwe-math-element"><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 },\|\cdot \|_{\infty }^{\mathrm {ess} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msup> <mo>,</mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mo>⋅<!-- ⋅ --></mo> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (L^{\infty },\|\cdot \|_{\infty }^{\mathrm {ess} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1da85f47ea44f56bb213504851b23c3d49484b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.564ex; height:2.843ex;" alt="{\displaystyle (L^{\infty },\|\cdot \|_{\infty }^{\mathrm {ess} })}"></span> till ett normerat rum med ekvivalensrelationen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼<!-- ∼ --></mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce5528a8c86c5b0121f9448aa9a117429f5b9c88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.195ex; height:1.343ex;" alt="{\displaystyle \sim \,}"></span> ovan. </p> <div class="mw-heading mw-heading3"><h3 id="Relation_till_andra_normer">Relation till andra normer</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supremumnormen&veaction=edit&section=5" title="Redigera avsnitt: Relation till andra normer" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Supremumnormen&action=edit&section=5" title="Redigera avsnitts källkod: Relation till andra normer"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Om f är en funktion så att <span class="mwe-math-element"><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\|_{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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}<\infty \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23096d8cbd8da744dd9de6e40ac99ca7c529e6a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.472ex; height:3.009ex;" alt="{\displaystyle \|f\|_{p}<\infty \,}"></span> och <span class="mwe-math-element"><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 }^{\text{ess}}<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>ess</mtext> </mrow> </msubsup> <mo><</mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }^{\text{ess}}<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01755428d19be75f456977db2bbf2529175a1a55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.284ex; height:2.843ex;" alt="{\displaystyle \|f\|_{\infty }^{\text{ess}}<\infty }"></span> så gäller att </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 \lim _{q\to \infty }\|f\|_{q}=\|f\|_{\infty }^{\text{ess}}}"> <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>q</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>q</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>ess</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{q\to \infty }\|f\|_{q}=\|f\|_{\infty }^{\text{ess}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12bd2c290313b3664fdebb651cd2c0cfa3ca306c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.982ex; height:4.176ex;" alt="{\displaystyle \lim _{q\to \infty }\|f\|_{q}=\|f\|_{\infty }^{\text{ess}}}"></span>.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Bevis">Bevis</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supremumnormen&veaction=edit&section=6" title="Redigera avsnitt: Bevis" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Supremumnormen&action=edit&section=6" title="Redigera avsnitts källkod: Bevis"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Låt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> vara större än <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>. </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\|_{q}=\left(\int |f|^{q}\right)^{1/q}=\left(\int |f|^{p}|f|^{q-p}\right)^{1/q}}"> <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>q</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>∫<!-- ∫ --></mo> <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>q</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>∫<!-- ∫ --></mo> <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> <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>q</mi> <mo>−<!-- − --></mo> <mi>p</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{q}=\left(\int |f|^{q}\right)^{1/q}=\left(\int |f|^{p}|f|^{q-p}\right)^{1/q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4bc89533083a10b3b2b43bcd2afc5d84119bb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.916ex; height:6.676ex;" alt="{\displaystyle \|f\|_{q}=\left(\int |f|^{q}\right)^{1/q}=\left(\int |f|^{p}|f|^{q-p}\right)^{1/q}}"></span></dd></dl> <p>Eftersom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q-p>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>−<!-- − --></mo> <mi>p</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q-p>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e35b1c2b65a4aca68cf2003aa9f080488428f118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.34ex; height:2.509ex;" alt="{\displaystyle q-p>0}"></span> är detta mindre än </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 \left(\int |f|^{p}\|f\|_{\infty }^{q-p}\right)^{1/q}=\|f\|_{\infty }^{1-p/q}\left(\int |f|^{p}\right)^{1/q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mo>∫<!-- ∫ --></mo> <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> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−<!-- − --></mo> <mi>p</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−<!-- − --></mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mo>∫<!-- ∫ --></mo> <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> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\int |f|^{p}\|f\|_{\infty }^{q-p}\right)^{1/q}=\|f\|_{\infty }^{1-p/q}\left(\int |f|^{p}\right)^{1/q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91462de6f687460b6d15bb3ba7f195ea655ea66f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.669ex; height:6.676ex;" alt="{\displaystyle \left(\int |f|^{p}\|f\|_{\infty }^{q-p}\right)^{1/q}=\|f\|_{\infty }^{1-p/q}\left(\int |f|^{p}\right)^{1/q}}"></span></dd></dl> <p>Eftersom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p/q>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−<!-- − --></mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-p/q>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5001c9da6a89c27c054c1be36870e151c32a7bd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.665ex; height:2.843ex;" alt="{\displaystyle 1-p/q>0}"></span> är detta mindre än </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 }\left(\int |f|^{p}\right)^{1/q}\to \|f\|_{\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 mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mo>∫<!-- ∫ --></mo> <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> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msup> <mo stretchy="false">→<!-- → --></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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }\left(\int |f|^{p}\right)^{1/q}\to \|f\|_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d1328f48d4cedab946eee98da1ecbb09511b817" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.838ex; height:6.676ex;" alt="{\displaystyle \|f\|_{\infty }\left(\int |f|^{p}\right)^{1/q}\to \|f\|_{\infty }}"></span> när <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\to \infty \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\to \infty \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1777529b4a132cd04b8d4d64595181b638f5b851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.394ex; height:2.176ex;" alt="{\displaystyle q\to \infty \,}"></span></dd></dl> <p>För den omvända olikheten, definiera <span class="mwe-math-element"><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=\{x|f(x)>a\}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\{x|f(x)>a\}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c14a21147be18d19b014523cfe21be5704ce8572" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.308ex; height:2.843ex;" alt="{\displaystyle E=\{x|f(x)>a\}\,}"></span>. Då är </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\|_{q}\geq \left(\int _{E}|f|^{q}\right)^{1/q}\geq a\mu (E)^{1/q}\to a}"> <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>q</mi> </mrow> </msub> <mo>≥<!-- ≥ --></mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</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>q</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msup> <mo>≥<!-- ≥ --></mo> <mi>a</mi> <mi>μ<!-- μ --></mi> <mo stretchy="false">(</mo> <mi>E</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msup> <mo stretchy="false">→<!-- → --></mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{q}\geq \left(\int _{E}|f|^{q}\right)^{1/q}\geq a\mu (E)^{1/q}\to a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf4db9a4b67abae06fc6e7fbd2e4d30e880c06e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.264ex; height:6.676ex;" alt="{\displaystyle \|f\|_{q}\geq \left(\int _{E}|f|^{q}\right)^{1/q}\geq a\mu (E)^{1/q}\to a}"></span> när <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\to \infty \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\to \infty \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1777529b4a132cd04b8d4d64595181b638f5b851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.394ex; height:2.176ex;" alt="{\displaystyle q\to \infty \,}"></span>.</dd></dl> <p>Detta gäller för alla <span class="mwe-math-element"><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<\|f\|_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<\|f\|_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c227a6a9b1d21f482cd6500a9d3c05dc8376bcf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.807ex; height:2.843ex;" alt="{\displaystyle a<\|f\|_{\infty }}"></span>. </p> <!-- NewPP limit report Parsed by mw‐web.eqiad.main‐67456955cd‐w5k7p Cached time: 20241121122514 Cache expiry: 2592000 Reduced expiry: false Complications: [show‐toc] CPU time usage: 0.077 seconds Real time usage: 0.186 seconds Preprocessor visited node count: 409/1000000 Post‐expand include size: 0/2097152 bytes Template argument size: 0/2097152 bytes Highest expansion depth: 2/100 Expensive parser function count: 0/500 Unstrip recursion depth: 0/20 Unstrip post‐expand size: 2808/5000000 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 0.000 1 -total --> <!-- Saved in parser cache with key svwiki:pcache:875263:|#|:idhash:canonical and timestamp 20241121122514 and revision id 29912858. 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