Kompaktheitssatz von Riesz

Der Kompaktheitssatz von Riesz ist ein Lehrsatz, welcher dem mathematischen Teilgebiet der Funktionalanalysis zuzurechnen ist. Er geht zurück auf den ungarischen Mathematiker Friedrich Riesz und gibt eine Charakterisierung derjenigen normierten $\mathbb {K}$ -Vektorräume ( $\mathbb {K} =\mathbb {R}$ oder $\mathbb {K} =\mathbb {C}$ ), welche endlich dimensional sind.^[1]^[2]

Formulierung des Satzes

Der Satz lässt sich formulieren wie folgt:^[1]

Ein normierter Vektorraum $X$ ist dann und nur dann endlichdimensional, wenn die abgeschlossene Einheitskugel in $X$ ein kompakter topologischer Unterraum ist.

Dabei kann der Satz gleichwertig auch wie folgt formuliert werden:^[2]

Ein normierter Vektorraum $X$ ist genau dann von endlicher Dimension, wenn in $X$ jede beschränkte Folge eine konvergente Teilfolge besitzt.

In der Herleitung des Satzes lässt sich der wesentliche Beweisschritt auf das Lemma von Riesz stützen.^[1]

Schärfere Version

Zum rieszschen Kompaktheitssatz gibt es die folgende schärfere Version, welche in der Monographie von Lutz Führer zu finden ist:^[3]

Sei $X$ ein separierter topologischer Vektorraum über $\mathbb {R}$ .

Dann sind gleichwertig:

(a) $X$ ist endlichdimensional.

(b) $X$ ist homöomorph zu einem $\mathbb {R} ^{n}\;\;(n\in \mathbb {N} _{0})$ .

(c) $X$ ist lokalkompakt.

Anmerkung

In der Einleitung und im Anhang der Monographie von Jürgen Appell und Martin Väth findet sich eine umfassende Liste von äquivalenten Bedingungen für die „Endlichdimensionalität“ normierter Räume.^[4]

Literatur

Jürgen Appell, Martin Väth: Elemente der Funktionalanalysis. Vektorräume, Operatoren und Fixpunktsätze. Vieweg Verlag, Wiesbaden 2005, ISBN 3-528-03222-7 (MR2371701).
Lutz Führer: Allgemeine Topologie mit Anwendungen. Vieweg Verlag, Braunschweig 1977, ISBN 3-528-03059-3.
Guido Walz [Red.]: Lexikon der Mathematik in sechs Bänden. Band 4. Spektrum Akademischer Verlag, Heidelberg / Berlin 2002, ISBN 3-8274-0436-3.

Einzelnachweise und Anmerkungen

↑ ^a ^b ^c Appell, Väth: Elemente der Funktionalanalysis. 2005, S. 38–41
↑ ^a ^b Lexikon der Mathematik. Band 4. 2002, S. 424.
↑ Führer: Allgemeine Topologie mit Anwendungen. 1977, S. 116–117.
↑ Jürgen Appell, Martin Väth: Elemente der Funktionalanalysis. Vektorräume, Operatoren und Fixpunktsätze. Vieweg Verlag, Wiesbaden 2005, ISBN 3-528-03222-7, S. 313–314 (MR2371701).

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