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<title>Orthogonalsystem</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Orthogonalsystem</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>In der <a href="Lineare_Algebra" title="Lineare Algebra">Linearen Algebra</a> und der <a href="Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a>, Teilgebieten der <a href="Mathematik" title="Mathematik">Mathematik</a>, ist ein <b>Orthogonalsystem</b> eine Menge von <a href="Vektor" title="Vektor">Vektoren</a> eines <a href="Vektorraum" title="Vektorraum">Vektorraums</a> mit <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukt</a> (<a href="Pr%C3%A4hilbertraum" title="Prähilbertraum">Prähilbertraum</a>), die paarweise aufeinander senkrecht stehen. Sind die Vektoren zusätzlich noch normiert (d. h., sie haben die <a href="Norm_(Mathematik)" title="Norm (Mathematik)">Norm</a> 1), so spricht man von einem <b>Orthonormalsystem</b>.

</p>

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

<p>Eine <a href="Teilmenge" title="Teilmenge">Teilmenge</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 M}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> eines <a href="Pr%C3%A4hilbertraum" title="Prähilbertraum">Prähilbertraums</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}">

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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> heißt Orthogonalsystem, wenn gilt:

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<ol><li>Je zwei verschiedene Vektoren aus <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 M}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> sind zueinander <a href="Orthogonalit%C3%A4t" title="Orthogonalität">orthogonal</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 \forall v,w\in M:v\neq w\Rightarrow \langle v,w\rangle =0}">

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<annotation encoding="application/x-tex">{\displaystyle \forall v,w\in M:v\neq w\Rightarrow \langle v,w\rangle =0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a0d3381144732df89db2c3dbe64c36f6cc11df1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.738ex; height:2.843ex;" alt="{\displaystyle \forall v,w\in M:v\neq w\Rightarrow \langle v,w\rangle =0}" loading="lazy"></span></li>

<li>Der <a href="Nullvektor" title="Nullvektor">Nullvektor</a> ist nicht in der Menge enthalten.</li></ol>

<p>Hier bezeichnet <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 v,w\rangle }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ced6f40b6f52b3d974786df295649b5c00e51708.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.635ex; height:2.843ex;" alt="{\displaystyle \langle v,w\rangle }" loading="lazy"></span> das <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukt</a> des Raums <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>

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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>, im euklidischen Raum also das <a href="Standardskalarprodukt" title="Standardskalarprodukt">Standardskalarprodukt</a>.

</p><p>Gilt zusätzlich

</p>

<dl><dd>Jeder Vektor aus <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 M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> ist normiert, d. h. <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 \forall v\in M:\langle v,v\rangle =1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall v\in M:\langle v,v\rangle =1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/077f0d1bf1b8e65ca2fdae42fc2a8ca9fca154b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19ex; height:2.843ex;" alt="{\displaystyle \forall v\in M:\langle v,v\rangle =1}" loading="lazy"></span>,</dd></dl>

<p>so nennt man <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 M}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> ein Orthonormalsystem.

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

<ul><li>Orthogonalsysteme sind <a href="Lineare_Unabh%C3%A4ngigkeit" title="Lineare Unabhängigkeit">linear unabhängig</a>.</li>

<li>In <a href="Separabler_Raum" title="Separabler Raum">separablen</a> <a href="Hilbertraum" title="Hilbertraum">Hilberträumen</a> (insbesondere in allen endlichdimensionalen Hilberträumen) lässt sich mit dem <a href="Gram-Schmidtsches_Orthogonalisierungsverfahren" title="Gram-Schmidtsches Orthogonalisierungsverfahren">Gram-Schmidtschen Orthogonalisierungsverfahren</a> aus jedem linear unabhängigen System ein Orthogonalsystem (bzw. Orthonormalsystem) bzw. aus jeder <a href="Schauderbasis" title="Schauderbasis">(Schauder-)Basis</a> eine orthogonale (bzw. orthonormale) Basis konstruieren.</li>

<li>Für ein Orthonormalsystem <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 M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> gilt die <a href="Besselsche_Ungleichung" title="Besselsche Ungleichung">Besselsche Ungleichung</a>

</mrow>

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<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{e\in M}|\langle x,\,e\rangle |^{2}\leq \|x\|^{2}\quad \forall ~x\in V.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6019bce98c60023c6ce91d7f1a2bb8e0ddf7769.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.575ex; height:5.676ex;" alt="{\displaystyle \sum _{e\in M}|\langle x,\,e\rangle |^{2}\leq \|x\|^{2}\quad \forall ~x\in V.}" loading="lazy"></span></dd></dl></li>

<li>Für jeden Vektor <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\in V}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa374e20b2db7f6b8caa71ff1865f7f84f215c9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.176ex;" alt="{\displaystyle x\in V}" loading="lazy"></span> ist die Menge 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 e\in M}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ef354bbee2ecdd82295a98cd2ec23ffbef54abc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.366ex; height:2.176ex;" alt="{\displaystyle e\in M}" loading="lazy"></span>, für 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 \langle x,e\rangle \neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle x,e\rangle \neq 0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/496e9539243131c5e1b05c07bf6e687262aa7524.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.517ex; height:2.843ex;" alt="{\displaystyle \langle x,e\rangle \neq 0}" loading="lazy"></span> gilt, höchstens abzählbar.</li></ul>

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

<msup>

</mrow>

</mrow>

</msup>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span> mit dem Standardskalarprodukt ist die <a href="Standardbasis" title="Standardbasis">Standardbasis</a> ein Orthogonalsystem</li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f3f1c02125a3b161aad32d469cda6a21baff682.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.431ex; height:3.176ex;" alt="{\displaystyle L^{2}([0,2\pi ])}" loading="lazy"></span> bilden die Funktionen <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 \cos(kx)}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c319bedf55a650c1a8f31d4f96c68d8a4ab33d9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.461ex; height:2.843ex;" alt="{\displaystyle \cos(kx)}" loading="lazy"></span> ein Orthogonalsystem (Siehe auch <a href="Trigonometrisches_Polynom" title="Trigonometrisches Polynom">trigonometrisches Polynom</a>)</li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

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</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> mit dem Skalarprodukt <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,b)\mapsto \sum a_{n}b_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a,b)\mapsto \sum a_{n}b_{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2bd49d69236263405a80d04fbca0064c80e73b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.091ex; height:3.843ex;" alt="{\displaystyle (a,b)\mapsto \sum a_{n}b_{n}}" loading="lazy"></span> bilden die Folgen <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,\cdots ,0,1,0,\cdots )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,\cdots ,0,1,0,\cdots )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c27f3d75853de531320a039b7a4037987c2be212.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.462ex; height:2.843ex;" alt="{\displaystyle (0,\cdots ,0,1,0,\cdots )}" loading="lazy"></span> ein Orthogonalsystem</li>

<li>In dem Prähilbertraum der Polynome mit Grad kleiner gleich 5, <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 {\mathcal {P}}^{5}([0,1])}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}^{5}([0,1])}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2edbe1447052e0b1c79efe13748788ff27af9f17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.246ex; height:3.176ex;" alt="{\displaystyle {\mathcal {P}}^{5}([0,1])}" loading="lazy"></span>, versehen mit dem <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>-Skalarprodukt <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,b)\mapsto \int _{0}^{1}ab}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a,b)\mapsto \int _{0}^{1}ab}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89d6591c778e77ea464002ecf4c7c4117d364093.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.818ex; height:6.176ex;" alt="{\displaystyle (a,b)\mapsto \int _{0}^{1}ab}" loading="lazy"></span>, bilden die Funktionen</li></ul>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e8204ad49e975d5a38dff1f463db664ec89a787.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.106ex; height:2.176ex;" alt="{\displaystyle x\mapsto 1}" 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 x\mapsto x-{\frac {1}{2}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\mapsto x-{\frac {1}{2}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85b451c116143e0276161ad8abe669d536ff44ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.112ex; height:5.176ex;" alt="{\displaystyle x\mapsto x-{\frac {1}{2}}}" loading="lazy"></span></dd></dl></dd>

<dd>ein Orthogonalsystem.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Orthonormalbasis" title="Orthonormalbasis">Vollständiges Orthonormalsystem</a></li></ul>

<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>: <cite style="font-style:italic">Funktionalanalysis</cite>. 6., korrigierte Auflage. Springer, Berlin 2007, ISBN 978-3-540-72533-6.<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:Orthogonalsystem&rft.au=Dirk+Werner&rft.btitle=Funktionalanalysis&rft.date=2007&rft.edition=6.%2C+korrigierte&rft.genre=book&rft.isbn=9783540725336&rft.place=Berlin&rft.pub=Springer" style="display:none"> </span> Kapitel V.3 (Für den unendlichdimensionalen Fall, dort finden sich auch Beweise für die Beispiele)</li>

<li>Gerd Fischer: <cite style="font-style:italic">Lineare Algebra: Eine Einführung für Studienanfänger</cite>. 13. Auflage. Vieweg, 2002, ISBN 3-528-97217-3.<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:Orthogonalsystem&rft.au=Gerd+Fischer&rft.btitle=Lineare+Algebra%3A+Eine+Einf%C3%BChrung+f%C3%BCr+Studienanf%C3%A4nger&rft.date=2002&rft.edition=13&rft.genre=book&rft.isbn=3528972173&rft.pub=Vieweg" style="display:none"> </span> (Für den endlichdimensionalen Fall, dort unter „Erzeugendensystem“)</li></ul></div><div><div class="zim-footer">

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