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<title>Operator endlichen Ranges</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Operator endlichen Ranges</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">Ein Operator endlichen Ranges oder endlichdimensionaler Operator ist ein <a href="Mathematik" title="Mathematik">mathematischer Begriff</a> aus der Theorie der <a href="Lineare_Abbildung" title="Lineare Abbildung">linearen Abbildungen</a>.

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

Ein Operator endlichen Ranges ist eine lineare Abbildung zwischen <a href="Vektorraum" title="Vektorraum">Vektorräumen</a>, so dass die <a href="Bildmenge" class="mw-redirect" title="Bildmenge">Bildmenge</a> ein <a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">endlichdimensionaler</a> Vektorraum ist.<a href="#cite_note-2">[1.1]</a><a href="#cite_note-Göpfert-3">[2]</a><a href="#cite_note-5">[3.1]</a>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon V\rightarrow W}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05a7957419733e7ddda97725b89d75619321d84a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.611ex; height:2.176ex;" alt="{\displaystyle F\colon V\rightarrow W}" loading="lazy"> eine lineare Abbildung zwischen zwei <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><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">-Vektorräumen, so ist die Bildmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Bild} (F):=\{F(v)\mid v\in V\}}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Bild} (F):=\{F(v)\mid v\in V\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a39f8c34ac9a599fc684d81f6793320e0d338291.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.222ex; height:2.843ex;" alt="{\displaystyle \operatorname {Bild} (F):=\{F(v)\mid v\in V\}}" loading="lazy"> ein <a href="Untervektorraum" title="Untervektorraum">Untervektorraum</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy">.

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> hat also endlichen Rang, wenn dieser Untervektorraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(V)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/224630ea52e5dbd55f96550a699ea53fa2654753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.337ex; height:2.843ex;" alt="{\displaystyle F(V)}" loading="lazy"> endlichdimensional ist.

Hierbei kann <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><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"> ein beliebiger <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> sein.

Bei Anwendungen, insbesondere in der <a href="Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a>, hat man es meistens mit den Körpern <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><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"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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

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</math><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"> der reellen beziehungsweise komplexen Zahlen zu tun.

Lineare Abbildungen nennt man oft auch lineare Operatoren. Die Dimension des Bildes nennt man den Rang des Operators.

Das entspricht dem aus der <a href="Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a> bekannten <a href="Rang_(Lineare_Algebra)" title="Rang (Lineare Algebra)">Rang</a> einer <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a>, denn bezüglich einer <a href="Basis_(Vektorraum)" title="Basis (Vektorraum)">Basis</a> kann man eine Matrix ja als lineare Abbildung auffassen.

Das erklärt die Bezeichnung Operator endlichen Ranges.

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

<ul><li>Ist mit obigen Bezeichnungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> endlichdimensional, so ist jeder lineare Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon V\rightarrow W}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05a7957419733e7ddda97725b89d75619321d84a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.611ex; height:2.176ex;" alt="{\displaystyle F\colon V\rightarrow W}" loading="lazy"> von endlichem Rang, denn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Bild} (F)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Bild} (F)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fa01eae04f090a1f7de12780f94b5697dca9fdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.782ex; height:2.843ex;" alt="{\displaystyle \operatorname {Bild} (F)}" loading="lazy"> muss als Untervektorraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> ebenfalls endlichdimensional sein.</li>

<li>Ist mit obigen Bezeichnungen <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><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"> endlichdimensional, so ist ebenfalls jeder lineare Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon V\rightarrow W}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05a7957419733e7ddda97725b89d75619321d84a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.611ex; height:2.176ex;" alt="{\displaystyle F\colon V\rightarrow W}" loading="lazy"> von endlichem Rang, denn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Bild} (F)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Bild} (F)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fa01eae04f090a1f7de12780f94b5697dca9fdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.782ex; height:2.843ex;" alt="{\displaystyle \operatorname {Bild} (F)}" loading="lazy"> kann keine größere Dimension als die von <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><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"> haben.</li>

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<annotation encoding="application/x-tex">{\displaystyle w_{1},\ldots w_{n}\in W}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/074c8a6f5d33cb2f9ea2d427c9764f9c5c4aae5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.021ex; height:2.509ex;" alt="{\displaystyle w_{1},\ldots w_{n}\in W}" loading="lazy"> und sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1},\ldots f_{n}\colon V\rightarrow K}">

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<annotation encoding="application/x-tex">{\displaystyle f_{1},\ldots f_{n}\colon V\rightarrow K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31791092c134fd85e3d6437c553ed98ab2439ddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.197ex; height:2.509ex;" alt="{\displaystyle f_{1},\ldots f_{n}\colon V\rightarrow K}" loading="lazy"> <a href="Lineares_Funktional" class="mw-redirect" title="Lineares Funktional">lineare Funktionale</a>, so ist</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle F\colon V\rightarrow W,\quad v\mapsto \sum _{i=0}f_{i}(v)w_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7707c36b80f5926b00682bf1cf6a244763a77c8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.791ex; height:5.509ex;" alt="{\displaystyle F\colon V\rightarrow W,\quad v\mapsto \sum _{i=0}f_{i}(v)w_{i}}" loading="lazy"></dd>

<dd>ein endlichdimensionaler Operator, denn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Bild} (F)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Bild} (F)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fa01eae04f090a1f7de12780f94b5697dca9fdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.782ex; height:2.843ex;" alt="{\displaystyle \operatorname {Bild} (F)}" loading="lazy"> ist definitionsgemäß in der <a href="Lineare_H%C3%BClle" title="Lineare Hülle">linearen Hülle</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1},\ldots w_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle w_{1},\ldots w_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ac5ea2a8d07c8eb3efe9fb05b7dcf24641d1f5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.745ex; height:2.009ex;" alt="{\displaystyle w_{1},\ldots w_{n}}" loading="lazy"> enthalten und daher endlichdimensional. Man kann zeigen, dass jeder endlichdimensionale Operator von dieser Form ist, sogar mit <a href="Lineare_Unabh%C3%A4ngigkeit" title="Lineare Unabhängigkeit">linear unabhängigen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1},\ldots w_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle w_{1},\ldots w_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ac5ea2a8d07c8eb3efe9fb05b7dcf24641d1f5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.745ex; height:2.009ex;" alt="{\displaystyle w_{1},\ldots w_{n}}" loading="lazy"> (dabei wird der <a href="Nulloperator" class="mw-redirect" title="Nulloperator">Nulloperator</a> durch die <a href="Leere_Summe" title="Leere Summe">leere Summe</a> dargestellt). Sind die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe22f0329d3ecb2e1880d44d191aba0e5475db68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle w_{i}}" loading="lazy"> linear unabhängig, so sind die Koeffizientenfunktionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65da883ca3d16b461e46c94777b0d9c4aa010e79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}" loading="lazy"> dadurch eindeutig bestimmt.<a href="#cite_note-6">[1.2]</a></dd></dl>

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

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</math><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"> unendlichdimensional, so ist der identische Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {id} _{V}\colon V\rightarrow V}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {id} _{V}\colon V\rightarrow V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50ce2f28724887485b9ded825406f26d8eb7ceea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.658ex; height:2.509ex;" alt="{\displaystyle \operatorname {id} _{V}\colon V\rightarrow V}" loading="lazy"> nicht von endlichem Rang, denn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Bild} (F)=V}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Bild} (F)=V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6b7cef174379fab6c3bb37d623a4fe9377abab3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.668ex; height:2.843ex;" alt="{\displaystyle \operatorname {Bild} (F)=V}" loading="lazy"> ist nicht endlichdimensional.</li></ul>

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

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

Hat man es mit <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><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">- oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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

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</math><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">-Vektorräumen mit einer <a href="Hausdorffraum" class="mw-redirect" title="Hausdorffraum">hausdorffschen</a> <a href="Topologischer_Vektorraum" title="Topologischer Vektorraum">Vektorraumtopologie</a> zu tun, so kann man die Frage nach der <a href="Stetige_Funktion#Stetigkeit_in_der_Topologie" title="Stetige Funktion">Stetigkeit</a> stellen.

Operatoren endlichen Ranges müssen nicht stetig sein, wie schon das Beispiel eines unstetigen linearen Funktionals zeigt.

Ist <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> ein endlichdimensionaler Operator der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle v\mapsto \sum _{i=0}^{n}f_{i}(v)w_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle v\mapsto \sum _{i=0}^{n}f_{i}(v)w_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31adef1cebabc6d000cc88c1eadcdce453a6273e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.822ex; height:3.176ex;" alt="{\displaystyle \textstyle v\mapsto \sum _{i=0}^{n}f_{i}(v)w_{i}}" loading="lazy"> mit linear unabhängigen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe22f0329d3ecb2e1880d44d191aba0e5475db68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle w_{i}}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> genau dann stetig, wenn die Koeffizientenfunktionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65da883ca3d16b461e46c94777b0d9c4aa010e79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}" loading="lazy"> stetig sind.<a href="#cite_note-7">[1.3]</a>

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

Stetige, endlichdimensionale Operatoren zwischen <a href="Normierter_Raum" title="Normierter Raum">normierten Räumen</a> sind <a href="Kompakter_Operator" title="Kompakter Operator">kompakt</a>,<a href="#cite_note-8">[1.4]</a><a href="#cite_note-9">[3.2]</a> denn das Bild der <a href="Einheitskugel" title="Einheitskugel">Einheitskugel</a> ist in einer Kugel eines endlichdimensionalen Raums enthalten und daher nach dem <a href="Satz_von_Heine-Borel" title="Satz von Heine-Borel">Satz von Heine-Borel</a> <a href="Relativ_kompakte_Teilmenge" title="Relativ kompakte Teilmenge">relativ kompakt</a>.

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

Summen und skalare Vielfache von Operatoren endlichen Ranges sind wieder von dieser Form.

Die Menge der Operatoren endlichen Ranges zwischen zwei Vektorräumen bildet als einen Vektorraum.

Hat man lineare Operatoren

</mrow>

</mpadded>

</mpadded>

</munderover>

</mrow>

</mrow>

</mpadded>

</mpadded>

</munderover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\quad {\xrightarrow[{}]{F}}\quad V\quad {\xrightarrow[{}]{G}}\quad W}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c896cff70fa1666eb003da2b20030c5fa06a67db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.654ex; margin-top: -0.293ex; margin-bottom: -0.518ex; width:20.465ex; height:5.509ex;" alt="{\displaystyle U\quad {\xrightarrow[{}]{F}}\quad V\quad {\xrightarrow[{}]{G}}\quad W}" loading="lazy"></dd></dl>

und ist einer von diesen endlichdimensional, so gilt das auch für ihre Komposition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\circ F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\circ F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9893b421c332c86e71c5279ba248b6ac40ce9711.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.762ex; height:2.176ex;" alt="{\displaystyle G\circ F}" loading="lazy">.

Die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}(V)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ab5d2dffef5df532e904f348022b9a3a4425d7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.908ex; height:2.843ex;" alt="{\displaystyle {\mathcal {E}}(V)}" loading="lazy"> der endlichdimensionalen Operatoren auf einem Vektorraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> bildet daher ein <a href="Ideal_(Mathematik)" class="mw-redirect" title="Ideal (Mathematik)">Ideal</a> in der <a href="Algebra_(Struktur)" class="mw-redirect" title="Algebra (Struktur)">Algebra</a> aller linearen Operatoren auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><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">. Aus demselben Grunde bildet die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(V)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ed650103ee7a63be048d8e2354b184d2a317160.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.523ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(V)}" loading="lazy"> der endlichdimensionalen, stetigen Operatoren auf einem <a href="Banachraum" title="Banachraum">Banachraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> ein Ideal in der <a href="Banachalgebra" title="Banachalgebra">Banachalgebra</a> der <a href="Beschr%C3%A4nkter_linearer_Operator" class="mw-redirect" title="Beschränkter linearer Operator">beschränkten linearen Operatoren</a>.<a href="#cite_note-10">[1.5]</a><a href="#cite_note-11">[3.3]</a>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> unendlichdimensional, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(V)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ed650103ee7a63be048d8e2354b184d2a317160.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.523ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(V)}" loading="lazy"> nicht <a href="Abgeschlossene_Menge" title="Abgeschlossene Menge">abgeschlossen</a> in der <a href="Operatortopologie" title="Operatortopologie">Operatortopologie</a>. Es ist im Ideal der <a href="Spurklasseoperator" title="Spurklasseoperator">Spurklasseoperatoren</a> und daher auch im Ideal der kompakten Operatoren enthalten. Hat <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> die <a href="Approximationseigenschaft" title="Approximationseigenschaft">Approximationseigenschaft</a>, so liegt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(V)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ed650103ee7a63be048d8e2354b184d2a317160.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.523ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(V)}" loading="lazy"> dicht im Ideal der kompakten Operatoren.

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

Zu linearen Operatoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon V\rightarrow W}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05a7957419733e7ddda97725b89d75619321d84a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.611ex; height:2.176ex;" alt="{\displaystyle F\colon V\rightarrow W}" loading="lazy"> kann man bekanntlich den <a href="Dualer_Operator" class="mw-redirect" title="Dualer Operator">dualen Operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{*}\colon W^{*}\rightarrow V^{*}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a216bbef5897383f99caaedf0545fb54e6e1c88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.05ex; height:2.343ex;" alt="{\displaystyle F^{*}\colon W^{*}\rightarrow V^{*}}" loading="lazy"> zwischen den <a href="Dualraum" title="Dualraum">Dualräumen</a> bilden.

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> von endlichem Rang, so auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8ab6d9b2b48fcfd2f5c741de789391d4cdd0f13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.869ex; height:2.343ex;" alt="{\displaystyle F^{*}}" loading="lazy">.

Hat <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> die Form

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(v)=\sum _{i=0}^{n}f_{i}(v)w_{i}\quad {\text{mit}}\quad f_{i}\in V^{*},w_{i}\in W,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd5559076f5e7c953565300cbde97388a5b8e238.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.162ex; height:6.843ex;" alt="{\displaystyle F(v)=\sum _{i=0}^{n}f_{i}(v)w_{i}\quad {\text{mit}}\quad f_{i}\in V^{*},w_{i}\in W,}" loading="lazy"></dd></dl>

so hat der duale Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{*}\colon W^{*}\rightarrow V^{*}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a216bbef5897383f99caaedf0545fb54e6e1c88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.05ex; height:2.343ex;" alt="{\displaystyle F^{*}\colon W^{*}\rightarrow V^{*}}" loading="lazy"> die Form

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F^{*}(g)=\sum _{i=0}^{n}{\hat {w_{i}}}(g)f_{i}\quad {\text{mit}}\quad f_{i}\in V^{*},w_{i}\in W,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60479526d0757a517d9107b25bd40f25ba5515a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.267ex; height:6.843ex;" alt="{\displaystyle F^{*}(g)=\sum _{i=0}^{n}{\hat {w_{i}}}(g)f_{i}\quad {\text{mit}}\quad f_{i}\in V^{*},w_{i}\in W,}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {w_{i}}}}">

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {w_{i}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f65474d3addcbad701cd83d0db533588f7096f78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.509ex;" alt="{\displaystyle {\hat {w_{i}}}}" loading="lazy"> das durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {w_{i}}}(g):=g(w_{i})}">

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {w_{i}}}(g):=g(w_{i})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acd357202b7d6a6461cbaa62f6febf082b057160.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.524ex; height:2.843ex;" alt="{\displaystyle {\hat {w_{i}}}(g):=g(w_{i})}" loading="lazy"> definierte lineare Funktional auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5292cb3dbf333bb0b51f39fb9f724fa8d4dfcc63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.562ex; height:2.343ex;" alt="{\displaystyle W^{*}}" loading="lazy"> ist.<a href="#cite_note-12">[1.6]</a>

Hat man es mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><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">- oder <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>

</semantics>

</math><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">-Vektorräumen mit einer <a href="Hausdorffraum" class="mw-redirect" title="Hausdorffraum">hausdorffschen</a> <a href="Topologischer_Vektorraum" title="Topologischer Vektorraum">Vektorraumtopologie</a> zu tun, so kann man zu stetigen, linearen Operatoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon V\rightarrow W}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05a7957419733e7ddda97725b89d75619321d84a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.611ex; height:2.176ex;" alt="{\displaystyle F\colon V\rightarrow W}" loading="lazy"> den stetigen, dualen Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F'\colon W'\rightarrow V'}">

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F'\colon W'\rightarrow V'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/102b7e0b00e5bc12f4a7cff994354fa3a114d624.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.942ex; height:2.509ex;" alt="{\displaystyle F'\colon W'\rightarrow V'}" loading="lazy"> bilden.

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> von endlichem Rang, so auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9cb2068c9fcb933aa21d1cfd42103514f5276e73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.509ex;" alt="{\displaystyle F'}" loading="lazy">.

Hat <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> obige Form, so erhält man für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9cb2068c9fcb933aa21d1cfd42103514f5276e73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.509ex;" alt="{\displaystyle F'}" loading="lazy"> dieselbe Formel wie für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8ab6d9b2b48fcfd2f5c741de789391d4cdd0f13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.869ex; height:2.343ex;" alt="{\displaystyle F^{*}}" loading="lazy">.<a href="#cite_note-12">[1.6]</a>

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

Sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> normierte Räume, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff31fe992a31b4c954a933f1be91e2739a1c0ae7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.602ex; height:2.509ex;" alt="{\displaystyle V'}" loading="lazy"> der Dualraum von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><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">, so ist die Abbildung

<msup>

</msup>

</mrow>

<msub>

</mrow>

</msub>

<mtext>wobei</mtext>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V'\times W\rightarrow {\mathcal {F}}(V,W),\quad (f,w)\mapsto F_{(f,w)},{\text{wobei}}\quad F_{(f,w)}(v):=f(v)w}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a97584eef8cd908551977aa6385b97945b1bf953.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:66.06ex; height:3.343ex;" alt="{\displaystyle V'\times W\rightarrow {\mathcal {F}}(V,W),\quad (f,w)\mapsto F_{(f,w)},{\text{wobei}}\quad F_{(f,w)}(v):=f(v)w}" loading="lazy"></dd></dl>

offenbar eine <a href="Bilineare_Abbildung" title="Bilineare Abbildung">bilineare Abbildung</a>, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(V,W)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(V,W)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1c1bd57b836b52bd12d022469f8f596f49ff469.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.992ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(V,W)}" loading="lazy"> der Raum der stetigen, linearen Operatoren endlichen Ranges ist. Die universelle Eigenschaft des <a href="Tensorprodukt" title="Tensorprodukt">Tensorprodukts</a> liefert einen <a href="Isomorphismus" title="Isomorphismus">Isomorphismus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V'\otimes W\cong {\mathcal {F}}(V,W)}">

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V'\otimes W\cong {\mathcal {F}}(V,W)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d80e79c52358f85fef1522dbd44fd85322201ac1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.968ex; height:3.009ex;" alt="{\displaystyle V'\otimes W\cong {\mathcal {F}}(V,W)}" loading="lazy">.<a href="#cite_note-13">[3.4]</a><a href="#cite_note-Ryan-14">[4]</a>

Der Raum der stetigen, linearen Operatoren endlichen Ranges kann also als Tensorprodukt geschrieben werden.

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

<li>↑

Harro Heuser: <cite style="font-style:italic">Funktionalanalysis</cite>. B. G. Teubner, Stuttgart 1975, ISBN 3-519-02206-0.<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:Operator+endlichen+Ranges&rft.au=Harro+Heuser&rft.btitle=Funktionalanalysis&rft.date=1975&rft.genre=book&rft.isbn=3519022060&rft.place=Stuttgart&rft.pub=B.+G.+Teubner" style="display:none">

<ol class="mw-subreference-list"><li id="cite_note-2"><a href="#cite_ref-2">↑</a> S. 67.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> S. 94.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> Satz 13.4

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> Satz 13.2

</li>

<li id="cite_note-10"><a href="#cite_ref-10">↑</a> S. 85.

</li>

<li id="cite_note-12">↑ <a href="#cite_ref-12">a</a> <a href="#cite_ref-12-1">b</a> Satz 17.1 für den Fall V=W

</li>

</ol></li>

Alfred Göpfert, Thomas Riedrich, Christiane Tammer: <cite style="font-style:italic">Angewandte Funktionalanalysis, Motivationen und Methoden für Mathematiker und Wirtschaftswissenschaftler</cite>. Vieweg & Teubner, 2009, ISBN 978-3-8351-0133-3, S. 52.<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:Operator+endlichen+Ranges&rft.au=Alfred+G%C3%B6pfert%2C+Thomas+Riedrich%2C+Christiane+Tammer&rft.btitle=Angewandte+Funktionalanalysis%2C+Motivationen+und+Methoden+f%C3%BCr+Mathematiker+und+Wirtschaftswissenschaftler&rft.date=2009-03-25&rft.genre=book&rft.isbn=9783835101333&rft.pages=52&rft.pub=Vieweg+%26+Teubner" style="display:none">

</li>

<li>↑

Y. A. Abramovich, Charalambos D. Aliprantis: <cite class="lang" lang="en" dir="auto" style="font-style:italic">An Invitation to Operator Theory</cite>. In: American Mathematical Society (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic">Graduate Studies in Mathematics</cite>. Band 50. Providence, Rhode Island 2002, ISBN 0-8218-2146-6, Kap 4.1 Finite-rank Operators (englisch).<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:Operator+endlichen+Ranges&rft.atitle=An+Invitation+to+Operator+Theory&rft.au=Y.+A.+Abramovich%2C+Charalambos+D.+Aliprantis&rft.btitle=Graduate+Studies+in+Mathematics&rft.date=2002-09-01&rft.genre=book&rft.isbn=0821821466&rft.place=Providence%2C+Rhode+Island&rft.volume=50" style="display:none">

<ol class="mw-subreference-list"><li id="cite_note-5"><a href="#cite_ref-5">↑</a> Definition 4.1

</li>

<li id="cite_note-9"><a href="#cite_ref-9">↑</a> Lemma 4.4

</li>

<li id="cite_note-11"><a href="#cite_ref-11">↑</a> S. 124.

</li>

<li id="cite_note-13"><a href="#cite_ref-13">↑</a> S. 125.

</li>

</ol></li>

Raymond A. Ryan: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Introduction to Tensor Products of Banach Spaces</cite>. Springer-Verlag, London, Berlin, Heidelberg 2002, ISBN 1-85233-437-1, S. 8 (englisch).<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:Operator+endlichen+Ranges&rft.au=Raymond+A.+Ryan&rft.btitle=Introduction+to+Tensor+Products+of+Banach+Spaces&rft.date=2002&rft.genre=book&rft.isbn=1852334371&rft.pages=8&rft.place=London%2C+Berlin%2C+Heidelberg&rft.pub=Springer-Verlag" style="display:none">

</li>

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