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<title>Linearform</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Linearform</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>Eine <b>Linearform</b> ist ein <a href="Mathematisches_Objekt" title="Mathematisches Objekt">Objekt</a> aus dem <a href="Teilgebiete_der_Mathematik" title="Teilgebiete der Mathematik">mathematischen Teilgebiet</a> der <a href="Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a>. Es handelt sich dabei um eine <a href="Lineare_Abbildung" title="Lineare Abbildung">lineare Abbildung</a> von einem <a href="Vektorraum" title="Vektorraum">Vektorraum</a> in den zugrundeliegenden <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a>.

</p><p>Im Kontext der <a href="Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a>, das heißt im Falle eines <a href="Topologischer_Vektorraum" title="Topologischer Vektorraum">topologischen</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 \mathbb {R} }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span>- oder <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 \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></span>-Vektorraums, sind die betrachteten Linearformen meistens <a href="Stetiges_lineares_Funktional" class="mw-redirect" title="Stetiges lineares Funktional">stetige lineare Funktionale</a>.

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

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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> ein Körper 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 V}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> ein <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}">

</mstyle>

</mrow>

<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>-Vektorraum. Eine Abbildung <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\colon V\to K}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d7cc113330b996cc1ab99f9e65495aa5304f483.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.78ex; height:2.509ex;" alt="{\displaystyle f\colon V\to K}" loading="lazy"></span> heißt <i>Linearform</i>, wenn für alle Vektoren <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 x,y\in V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y\in V}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89e1c8106b3df898a0338c2ebce32902e4941fc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.147ex; height:2.509ex;" alt="{\displaystyle x,y\in V}" loading="lazy"></span> und Skalare <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 \alpha \in K}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3581f6410248bed60440badc5bd362ca1cc8c82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.394ex; height:2.176ex;" alt="{\displaystyle \alpha \in K}" loading="lazy"></span> gilt:

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</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11e072f8427aa606b95bad4d8fba9cb3da2c0b09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.013ex; height:2.843ex;" alt="{\displaystyle f(x+y)=f(x)+f(y)}" loading="lazy"></span> (Additivität);</li>

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c94613df5ebe92d192c6ac41572dd5e821474210.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.909ex; height:2.843ex;" alt="{\displaystyle f(\alpha x)=\alpha f(x)}" loading="lazy"></span> (Homogenität).</li></ol>

<p>Die Menge aller Linearformen über einem gegebenen Vektorraum <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 V}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> bildet dessen <a href="Dualraum" title="Dualraum">Dualraum</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 V^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5910e6a94f4f7ee2ee85ceed9dacef3eff7a6242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle V^{*}}" loading="lazy"></span> und damit selbst wieder in natürlicher Weise einen <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}">

</mstyle>

</mrow>

<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>-Vektorraum.

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

<p>Allgemeine Eigenschaften für Linearformen sind zum Beispiel:

</p>

<ul><li>Wie jede lineare Abbildung sind sie durch ihre Werte für eine beliebige <a href="Basis_(Vektorraum)" title="Basis (Vektorraum)">Basis</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 V}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> vollständig bestimmt.</li>

<li>Sie sind entweder trivial (überall identisch <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 0_{K}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0_{K}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eb8a92b2536158b176450bd0e3d43043539c0c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.856ex; height:2.509ex;" alt="{\displaystyle 0_{K}}" loading="lazy"></span>) oder <a href="Surjektivit%C3%A4t" class="mw-redirect" title="Surjektivität">surjektiv</a>.</li>

<li>Haben zwei von ihnen gleiche <a href="Kern_(Algebra)" title="Kern (Algebra)">Kerne</a>, so unterscheiden sie sich nur durch die <a href="Skalarmultiplikation" title="Skalarmultiplikation">Multiplikation</a> mit einem Skalar.</li></ul>

<p>Speziell für lineare Funktionale gilt außerdem:

</p>

<ul><li>Sie sind genau dann <a href="Stetige_Funktion" title="Stetige Funktion">stetig</a> wenn ihr Kern <a href="Abgeschlossene_Menge" title="Abgeschlossene Menge">abgeschlossen</a> ist.</li>

<li>Ihr <a href="Absoluter_Betrag" class="mw-redirect" title="Absoluter Betrag">absoluter Betrag</a> ist stets eine <a href="Halbnorm" title="Halbnorm">Halbnorm</a> auf <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 V}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span>.</li>

<li>Lineare Funktionale <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 \mathbb {K} ^{n}\to \mathbb {K} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a34964924b6012d921d3cf929d53929172b5bf0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.449ex; height:2.343ex;" alt="{\displaystyle \mathbb {K} ^{n}\to \mathbb {K} }" loading="lazy"></span> sind genau die Abbildungen <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 x\mapsto \langle v,x\rangle }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\mapsto \langle v,x\rangle }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f346a1621285ed48b77d0d061c32dbdd6da24905.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.244ex; height:2.843ex;" alt="{\displaystyle x\mapsto \langle v,x\rangle }" loading="lazy"></span>, 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 v\in \mathbb {K} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\in \mathbb {K} ^{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02e36cea802fa0eb9e4e2151644cba8947c9873a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.995ex; height:2.343ex;" alt="{\displaystyle v\in \mathbb {K} ^{n}}" loading="lazy"></span> einen Vektor 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 \langle \cdot ,\cdot \rangle }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a50080b735975d8001c9552ac2134b49ad534c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }" loading="lazy"></span> das <a href="Standardskalarprodukt" title="Standardskalarprodukt">Standardskalarprodukt</a> bezeichnen.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Linearform_als_Tensor">Linearform als Tensor</h2></div>

<p>Eine Linearform <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> ist ein <a href="Kontravarianz_(Physik)" class="mw-redirect" title="Kontravarianz (Physik)">kovarianter</a> <a href="Tensor" title="Tensor">Tensor</a> erster Stufe; man nennt sie deshalb manchmal auch <a href="1-Form" class="mw-redirect" title="1-Form">1-Form</a>. 1-Formen bilden die Grundlage für die Einführung von <a href="Differentialform" title="Differentialform">Differentialformen</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Verwandte_Begriffe">Verwandte Begriffe</h2></div>

<p>Gilt speziell <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=\mathbb {C} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58665cdd4df26adaa88a248908d1481041a77c9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.842ex; height:2.176ex;" alt="{\displaystyle K=\mathbb {C} }" loading="lazy"></span> und ändert man die zweite Bedingung 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 f(\alpha x)={\overline {\alpha }}f(x)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc00ddf6995f0172557d42519980a52fc65c3ef8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.024ex; height:2.843ex;" alt="{\displaystyle f(\alpha x)={\overline {\alpha }}f(x)}" loading="lazy"></span> ab, 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 {\overline {\alpha }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f87618f0333926d073474c3e9b99c65628f6c09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.603ex; height:2.343ex;" alt="{\displaystyle {\overline {\alpha }}}" loading="lazy"></span> das <a href="Komplexe_Konjugation" title="Komplexe Konjugation">komplex Konjugierte</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 \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> bezeichnet, erhält man eine <a href="Semilinearform" class="mw-redirect" title="Semilinearform">Semilinearform</a>.

</p><p>Eine Abbildung, die linear oder semilinear in mehr als einem Argument ist, ist eine <a href="Sesquilinearform" title="Sesquilinearform">Sesquilinearform</a>, eine <a href="Bilinearform" title="Bilinearform">Bilinearform</a>, oder allgemein eine <a href="Multilinearform" title="Multilinearform">Multilinearform</a>.

</p>

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

<ul><li><cite class="lang" lang="en" dir="auto" style="font-style:italic">Linear form</cite>. In: <a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a> (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="Encyclopedia_of_Mathematics" class="mw-redirect" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></cite>. Springer-Verlag und <a href="European_Mathematical_Society" title="European Mathematical Society">EMS</a> Press, Berlin 2002, ISBN 1-55608-010-7 (englisch, <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/linear_form">encyclopediaofmath.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Linearform&rft.atitle=Linear+form&rft.btitle=Encyclopedia+of+Mathematics&rft.date=2002&rft.genre=book&rft.isbn=1556080107&rft.place=Berlin&rft.pub=Springer-Verlag+und+EMS+Press" style="display:none"> </span></li></ul>

<ul><li><a href="Gerd_Fischer_(Mathematiker)" title="Gerd Fischer (Mathematiker)">Gerd Fischer</a>: <cite style="font-style:italic">Lineare Algebra</cite>. Eine Einführung für Studienanfänger. 16., überarbeitete und erweiterte Auflage. <a href="Vieweg%2BTeubner" class="mw-redirect" title="Vieweg+Teubner">Vieweg+Teubner</a>, Wiesbaden 2008, ISBN 978-3-8348-0428-0, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>280–281</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Linearform&rft.au=Gerd+Fischer&rft.btitle=Lineare+Algebra&rft.date=2008&rft.edition=16.%2C+%C3%BCberarbeitete+und+erweiterte&rft.genre=book&rft.isbn=9783834804280&rft.pages=280-281&rft.place=Wiesbaden&rft.pub=Vieweg%2BTeubner" style="display:none"> </span></li>

<li><a href="Harro_Heuser" title="Harro Heuser">Harro Heuser</a>: <cite style="font-style:italic">Funktionalanalysis</cite>. Theorie und Anwendung (= <cite style="font-style:italic">Mathematische Leitfäden</cite>). 4. Auflage. <a href="B._G._Teubner" class="mw-redirect" title="B. G. Teubner">B. G. Teubner</a>, Wiesbaden 2006, ISBN 978-3-8351-0026-8 (<a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=RT&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Heuser%2C%20Harro&s5=&s6=&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=3&mx-pid=2380292">MR2380292</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Linearform&rft.au=Harro+Heuser&rft.btitle=Funktionalanalysis&rft.date=2006&rft.edition=4.&rft.genre=book&rft.isbn=9783835100268&rft.place=Wiesbaden&rft.pub=B.+G.+Teubner&rft.series=Mathematische+Leitf%C3%A4den" style="display:none"> </span></li>

<li>Eberhard Oeljeklaus, <a href="Reinhold_Remmert" title="Reinhold Remmert">Reinhold Remmert</a>: <cite style="font-style:italic">Lineare Algebra I</cite> (= <cite style="font-style:italic">Heidelberger Taschenbücher</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>150</span>). Springer Verlag, Berlin (u. a.) 1974, ISBN 3-540-06715-9 (<a rel="nofollow" class="external text" href="http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Oeljeklaus&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=44&mx-pid=366944">MR0366944</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Linearform&rft.au=Eberhard+Oeljeklaus%2C+Reinhold+Remmert&rft.btitle=Lineare+Algebra+I&rft.date=1974&rft.genre=book&rft.isbn=3540067159&rft.place=Berlin+%28u.+a.%29&rft.pub=Springer+Verlag&rft.series=Heidelberger+Taschenb%C3%BCcher" style="display:none"> </span></li>

<li><a href="Walter_Rudin" title="Walter Rudin">Walter Rudin</a>: <i>Functional Analysis</i>, 2nd Ed., McGraw-Hill Inc., New York, 1991</li>

<li><a href="Dirk_Werner_(Mathematiker)" title="Dirk Werner (Mathematiker)">Dirk Werner</a>: <cite style="font-style:italic">Funktionalanalysis</cite>. 5., erw. Auflage. Springer-Verlag Berlin Heidelberg, Berlin 2005, ISBN 3-540-21381-3.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Linearform&rft.au=Dirk+Werner&rft.btitle=Funktionalanalysis&rft.date=2005&rft.genre=book&rft.isbn=3540213813&rft.place=Berlin&rft.pub=Springer-Verlag+Berlin+Heidelberg" style="display:none"> </span></li></ul></div><div><div class="zim-footer">

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