Riesz compactness theorem

The compact set of Riesz is a theorem , which is the mathematical sub-region of the functional analysis is attributable. It goes back to the Hungarian mathematician Friedrich Riesz and gives a characterization of those normalized - vector spaces ( or ) which are finite dimensional . ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {K} =\mathbb {R} }$ ${\displaystyle \mathbb {K} =\mathbb {C} }$

Formulation of the sentence

The sentence can be formulated as follows:

A normalized vector space is finite dimensional if and only if the closed unit sphere is in a compact topological subspace . ${\displaystyle X}$ ${\displaystyle X}$

The sentence can also be formulated as follows:

A normalized vector space is of finite dimension if and only if every bounded sequence has a convergent subsequence . ${\displaystyle X}$ ${\displaystyle X}$

In the derivation of the theorem, the essential proof step can be based on Riesz's lemma .

Sharper version

There is the following sharper version of the Riesz compactness theorem, which can be found in the monograph by Lutz Führer :

Let be a separated topological vector space over . ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$

Then are equivalent:

(a) is finite dimensional. ${\displaystyle X}$

(b) is homeomorphic to one . ${\displaystyle X}$ ${\displaystyle \mathbb {R} ^{n}\;\;(n\in \mathbb {N} _{0})}$

(c) is locally compact. ${\displaystyle X}$

annotation

In the introduction and in the appendix of the monograph by Jürgen Appell and Martin Väth there is a comprehensive list of equivalent conditions for the “finite dimensionality” of standardized spaces.

literature

• Jürgen Appell, Martin Väth : Elements of functional analysis . Vector spaces, operators and fixed point sets. Vieweg Verlag, Wiesbaden 2005, ISBN 3-528-03222-7 ( MR2371701 ).
• Lutz Führer: General topology with applications . Vieweg Verlag, Braunschweig 1977, ISBN 3-528-03059-3 .
• Guido Walz [Red.]: Lexicon of Mathematics in six volumes . tape 4 . Spectrum Academic Publishing House, Heidelberg / Berlin 2002, ISBN 3-8274-0436-3 .

References and comments

1. ↑ ^{a b c} Appell, Väth: Elements of functional analysis. 2005, pp. 38-41
2. ↑ ^{a b} Lexicon of Mathematics . Volume 4. 2002, p. 424.
3. ↑ Guide: General Topology with Applications. 1977, pp. 116-117.
4. ↑ Jürgen Appell, Martin Väth: Elements of functional analysis . Vector spaces, operators and fixed point sets. Vieweg Verlag, Wiesbaden 2005, ISBN 3-528-03222-7 , p. 313-314 ( MR2371701 ).