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<title>Satz von Fischer-Riesz</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Satz von Fischer-Riesz</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Der <b>Satz von Fischer-Riesz</b> ist eine Aussage aus der <a href="Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a>. <a href="Ernst_Sigismund_Fischer" title="Ernst Sigismund Fischer">Ernst Sigismund Fischer</a> und <a href="Frigyes_Riesz" title="Frigyes Riesz">Frigyes Riesz</a> bewiesen im Jahr 1907<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> unabhängig voneinander diesen Satz. Aus diesem Grund trägt die Aussage ihre Namen. In der Literatur finden sich heute unterschiedliche Sätze, die ihren Namen tragen und zum Teil Verallgemeinerungen dieses Satzes sind.

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<div class="mw-heading mw-heading2"><h2 id="Klassischer_Satz_von_Fischer-Riesz">Klassischer Satz von Fischer-Riesz</h2></div>

<p>Fischer und Riesz bewiesen die folgende Aussage. Der Raum <span class="mwe-math-element mwe-math-element-inline"><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^{2}([0,1])}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}([0,1])}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13c2586b254365ca72d662d9e6485fdee8026912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.099ex; height:3.176ex;" alt="{\displaystyle L^{2}([0,1])}" loading="lazy"></span> der quadrat-integrierbaren Funktionen ist <a href="Isometrie" title="Isometrie">isometrisch</a> <a href="Isomorphie_(Mathematik)" class="mw-redirect" title="Isomorphie (Mathematik)">isomorph</a> zum <a href="Folgenraum" title="Folgenraum">Folgenraum</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}(\mathbb {N} )}">

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<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(\mathbb {N} )}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30f5da85c0ac82572e622d7068b8cf4d0ac98fa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.511ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}(\mathbb {N} )}" loading="lazy"></span> der quadrat-summierbaren Funktionen also

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<annotation encoding="application/x-tex">{\displaystyle L^{2}([0,1])\cong \ell ^{2}(\mathbb {N} ).}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55469c0a65e877927b63b6f69dc3fa4f2de3c77b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.356ex; height:3.176ex;" alt="{\displaystyle L^{2}([0,1])\cong \ell ^{2}(\mathbb {N} ).}" loading="lazy"></span></dd></dl>

<p>Dies kann man auch weniger abstrakt in der Sprache der <a href="Analysis" title="Analysis">reellen Analysis</a> formulieren. So ist eine <a href="Messbare_Funktion" title="Messbare Funktion">messbare Funktion</a> genau dann in <span class="mwe-math-element mwe-math-element-inline"><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^{2}([-\pi ,\pi ])}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}([-\pi ,\pi ])}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f84fea7a212acaf14649b6cdcca282b0646a8b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.246ex; height:3.176ex;" alt="{\displaystyle L^{2}([-\pi ,\pi ])}" loading="lazy"></span>, wenn ihre <a href="Fourierreihe" title="Fourierreihe">Fourier-Reihe</a> bezüglich der <span class="mwe-math-element mwe-math-element-inline"><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^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy"></span>-Norm konvergiert. Im Folgenden wird der <span class="mwe-math-element mwe-math-element-inline"><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^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy"></span>-Raum von dem Intervall <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-\pi ,\pi ]}">

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<annotation encoding="application/x-tex">{\displaystyle [-\pi ,\pi ]}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb064fd6c55820cfa660eabeeda0f6e3c4935ae6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.8ex; height:2.843ex;" alt="{\displaystyle [-\pi ,\pi ]}" loading="lazy"></span> gebildet, dies erspart Normierungen, jedoch ist die Aussage auch für alle anderen kompakten Intervalle richtig.

</p><p>Die am <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span>-ten Glied abgebrochene Fourier-Reihe einer quadrat-integrierbaren Funktion <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></span> ist

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{N}(f)(x)=\sum _{n=-N}^{N}a_{n}\,\mathrm {e} ^{inx},}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06f27b5edb5e51407b14b91c9bba8e712fe13031.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.318ex; height:7.509ex;" alt="{\displaystyle {\mathcal {F}}_{N}(f)(x)=\sum _{n=-N}^{N}a_{n}\,\mathrm {e} ^{inx},}" loading="lazy"></span></dd></dl>

<p>wobei <span class="mwe-math-element mwe-math-element-inline"><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_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/790f9209748c2dca7ed7b81932c37c02af1dbc31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}" loading="lazy"></span> der n-te Koeffizient der Reihe ist, welche durch

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<annotation encoding="application/x-tex">{\displaystyle a_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)\,\mathrm {e} ^{-inx}\,\mathrm {d} x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b798f65db76b200b74a7c5bf922f9a9a3d4c3f02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.248ex; height:6.009ex;" alt="{\displaystyle a_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)\,\mathrm {e} ^{-inx}\,\mathrm {d} x}" loading="lazy"></span></dd></dl>

<p>gegeben ist. Für eine quadrat-integrierbare Funktion <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></span> gilt also dann

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<annotation encoding="application/x-tex">{\displaystyle \lim _{N\to \infty }\left\Vert {\mathcal {F}}_{N}(f)-f\right\|_{L^{2}}=\lim _{N\to \infty }\left\Vert \sum _{n=-N}^{N}a_{n}\,\mathrm {e} ^{in\cdot }-f\right\|_{L^{2}}=0.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/590fed04c0a9ebf9148c3db15dd2124d85c74c87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:53.476ex; height:7.676ex;" alt="{\displaystyle \lim _{N\to \infty }\left\Vert {\mathcal {F}}_{N}(f)-f\right\|_{L^{2}}=\lim _{N\to \infty }\left\Vert \sum _{n=-N}^{N}a_{n}\,\mathrm {e} ^{in\cdot }-f\right\|_{L^{2}}=0.}" loading="lazy"></span></dd></dl>

<p>Der Isomorphismus zwischen <span class="mwe-math-element mwe-math-element-inline"><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^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}(\mathbb {N} )}">

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<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(\mathbb {N} )}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30f5da85c0ac82572e622d7068b8cf4d0ac98fa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.511ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}(\mathbb {N} )}" loading="lazy"></span> ist also die Transformation in eine Fourier-Reihe.

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<div class="mw-heading mw-heading2"><h2 id="Verallgemeinerter_Satz_von_Fischer-Riesz">Verallgemeinerter Satz von Fischer-Riesz</h2></div>

<p>Oftmals findet man auch folgende, allgemeinere Aussage unter dem Namen Satz von Fischer-Riesz.

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<div class="mw-heading mw-heading3"><h3 id="Aussage">Aussage</h3></div>

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<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> ein <a href="Hilbertraum" title="Hilbertraum">Hilbertraum</a> und <span class="mwe-math-element mwe-math-element-inline"><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(e_{i}\right)_{i\in I}}">

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<annotation encoding="application/x-tex">{\displaystyle \left(e_{i}\right)_{i\in I}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0c357b43c962984a0735e72125e6c53a9eac318.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.417ex; height:3.009ex;" alt="{\displaystyle \left(e_{i}\right)_{i\in I}}" loading="lazy"></span> eine <a href="Orthonormalbasis" title="Orthonormalbasis">Orthonormalbasis</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span>, so ist die <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Abbildung</a>

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<annotation encoding="application/x-tex">{\displaystyle \Phi \colon \,H\to \ell ^{2}(I);\quad x\mapsto \left(\langle x,e_{i}\rangle \right)_{i\in I}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c843bc96d97b216fe6fb421157cd799d2db22d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.672ex; height:3.343ex;" alt="{\displaystyle \Phi \colon \,H\to \ell ^{2}(I);\quad x\mapsto \left(\langle x,e_{i}\rangle \right)_{i\in I}}" loading="lazy"></span></dd></dl>

<p>ein <a href="Isometrie" title="Isometrie">isometrischer</a> <a href="Isomorphismus" title="Isomorphismus">Isomorphismus</a>.

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<div class="mw-heading mw-heading3"><h3 id="Folgerungen">Folgerungen</h3></div>

<ul><li>Seien <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"></span> zwei Indexmengen. Zwei Hilberträume <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> mit Orthonormalbasen <span class="mwe-math-element mwe-math-element-inline"><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(e_{i}\right)_{i\in I}}">

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<annotation encoding="application/x-tex">{\displaystyle \left(e_{i}\right)_{i\in I}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0c357b43c962984a0735e72125e6c53a9eac318.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.417ex; height:3.009ex;" alt="{\displaystyle \left(e_{i}\right)_{i\in I}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><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(e_{j}\right)_{j\in J}}">

<msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(e_{j}\right)_{j\in J}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46a887b69279204c3e38606e27268b49c9847047.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.849ex; height:3.343ex;" alt="{\displaystyle \left(e_{j}\right)_{j\in J}}" loading="lazy"></span> sind isometrisch isomorph, wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"></span> die gleiche <a href="M%C3%A4chtigkeit_(Mathematik)" title="Mächtigkeit (Mathematik)">Kardinalität</a> haben.</li></ul>

<ul><li>Jedes Orthonormalsystem in einem Hilbertraum kann zu einer Orthonormalbasis ergänzt werden (was sich unmittelbar aus dem <a href="Lemma_von_Zorn" title="Lemma von Zorn">Lemma von Zorn</a> ergibt), insbesondere besitzt jeder Hilbertraum, da die <a href="Leere_Menge" title="Leere Menge">leere Menge</a> stets ein Orthonormalsystem ist, eine Orthonormalbasis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span>. Somit ist nach dem Satz von Fischer-Riesz jeder Hilbertraum isomorph zum Raum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}(B)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(B)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ec10dbcecdcb7ede7aadbea8f75b99109e08705.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.597ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}(B)}" loading="lazy"></span>.</li></ul>

<ul><li>Anders ausgedrückt: Die volle <a href="Unterkategorie" class="mw-redirect" title="Unterkategorie">Unterkategorie</a> der Räume <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}(I)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(I)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da952c66063a82ac4fb47473accc8669da064622.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.005ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}(I)}" loading="lazy"></span> für beliebige Mengen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"></span> in der Kategorie der Hilberträume mit geeigneten Morphismen (lineare Operatoren, beschränkte lineare Operatoren, lineare Kontraktionen) ist äquivalent zu dieser.</li></ul>

<ul><li>Aus dem Satz lässt sich folgern, dass jeder <a href="Separabler_Raum" title="Separabler Raum">separable</a> <a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">unendlichdimensionale</a> Hilbertraum zum <a href="Folgenraum" title="Folgenraum">Folgenraum</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}(\mathbb {N} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(\mathbb {N} )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30f5da85c0ac82572e622d7068b8cf4d0ac98fa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.511ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}(\mathbb {N} )}" loading="lazy"></span> isometrisch isomorph ist.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Vollständigkeit_der_Lp-Räume"><span id="Vollst.C3.A4ndigkeit_der_Lp-R.C3.A4ume"></span>Vollständigkeit der L<sup>p</sup>-Räume</h2></div>

<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Dualit%C3%A4t_von_Lp-R%C3%A4umen" title="Dualität von Lp-Räumen">Dualität von <span class="mwe-math-element mwe-math-element-inline"><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^{p}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf2317aaca1ecee4b8ccf667bc1001059eae5850.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}" loading="lazy"></span>-Räumen</a></i></div>

<p>Die Aussage, dass die <span class="mwe-math-element mwe-math-element-inline"><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^{p}(\Omega ,\mu )}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{p}(\Omega ,\mu )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5075016f83493f7cded328bb0a03a64b91e1cfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.565ex; height:2.843ex;" alt="{\displaystyle L^{p}(\Omega ,\mu )}" loading="lazy"></span>-Räume für <span class="mwe-math-element mwe-math-element-inline"><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\leq p\leq \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq \infty }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7d63bb8c8def80f4fb709fbb2aae6a11c9cd41d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq \infty }" loading="lazy"></span> mit der Norm

</p>

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}:=\left(\int _{\Omega }|f(x)|^{p}\,\mathrm {d} \mu (x)\right)^{1/p}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c53ad761e75723b1824e33726fc56e67484df037.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.621ex; height:6.676ex;" alt="{\displaystyle \|f\|_{p}:=\left(\int _{\Omega }|f(x)|^{p}\,\mathrm {d} \mu (x)\right)^{1/p}}" loading="lazy"></span></dd></dl>

<p><a href="Banachraum" title="Banachraum">Banachräume</a>, also insbesondere <a href="Vollst%C3%A4ndiger_Raum" title="Vollständiger Raum">vollständig</a> sind, wird auch oftmals als Satz von Fischer-Riesz bezeichnet.

</p><p>Für den Fall <span class="mwe-math-element mwe-math-element-inline"><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=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p=2}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d62e4100b94c1939c67f2d4b8580d26c78106c44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=2}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span> als <a href="Lebesgue-Ma%C3%9F" title="Lebesgue-Maß">Lebesgue-Maß</a> folgt dies nämlich aus dem Beweis des (klassischen) Satzes von Fischer-Riesz. So konvergiert die Folge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle a_{n}:=\int _{\Omega }f(x)\,\mathrm {e} ^{-inx}\,\mathrm {d} \mu (x)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle a_{n}:=\int _{\Omega }f(x)\,\mathrm {e} ^{-inx}\,\mathrm {d} \mu (x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0459eefb0a5946542c95be4bc2336cd87b9b5b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.159ex; height:3.176ex;" alt="{\displaystyle \textstyle a_{n}:=\int _{\Omega }f(x)\,\mathrm {e} ^{-inx}\,\mathrm {d} \mu (x)}" loading="lazy"></span> genau dann in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91f1f909abd70bd3d8fff0f7ae1ac23052387e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}}" loading="lazy"></span>, wenn <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></span> eine <span class="mwe-math-element mwe-math-element-inline"><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^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy"></span>-Funktion ist.

</p><p>Für <span class="mwe-math-element mwe-math-element-inline"><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<\infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1<p<\infty }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5540fd86346b4798d7447b8de70fe98cf6243d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1<p<\infty }" loading="lazy"></span> ergibt sich die Vollständigkeit des <span class="mwe-math-element mwe-math-element-inline"><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^{p}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf2317aaca1ecee4b8ccf667bc1001059eae5850.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}" loading="lazy"></span>-Raumes beispielsweise wegen dessen <a href="Reflexiver_Raum" title="Reflexiver Raum">Reflexivität</a>, die aus der <a href="Dualit%C3%A4t_von_Lp-R%C3%A4umen" title="Dualität von Lp-Räumen">Dualität von Lp-Räumen</a> resultiert. Jeder reflexive <a href="Normierter_Raum" title="Normierter Raum">normierte Raum</a> ist ein Banachraum, denn er ist nach Definition isomorph zum vollständigen <a href="Bidualraum" class="mw-redirect" title="Bidualraum">Bidualraum</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Sur les systèmes orthogonaux de fonctions, C. R. Paris 144 (1907) 615-619</span>

</li>

</ol>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Dirk_Werner_(Mathematiker)" title="Dirk Werner (Mathematiker)">Dirk Werner</a>: <i>Funktionalanalysis</i>, Springer-Verlag, Berlin, 2007, ISBN 978-3-540-72533-6</li>

<li>H. Heuser: <i>Funktionalanalysis</i>, Teubner-Verlag, 2006, ISBN 3-8351-0026-2, enthält historische Bemerkungen</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-06-04" href="https://de.wikipedia.org/wiki/?title=Satz_von_Fischer-Riesz&oldid=256695839">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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