Banach-Stone's theorem
The set of Banach-Stone is a classic mathematical theorem , which in the transition field between topology and functional analysis is based. It goes back to the two mathematicians Stefan Banach and Marshall Stone . The statement of the theorem can be summarized in such a way that the structure of a compact Hausdorff space and the structure of the associated Banach space of the continuous real-valued functions given on it are directly linked to each other and mutually determine each other except for isomorphism .
The Banach-Stone theorem is the subject of investigation and the starting point for a series of further investigations.
Formulation of the sentence
Given are two compact Hausdorff spaces and , in addition , the associated -anach spaces of the continuous real functions and , each provided with the supremum norm .
Then: and are homeomorphic if and only if and as Banach spaces are isometrically isomorphic .
annotation
The statement of the theorem is correct in the same way if one considers the corresponding function banach spaces of the continuous complex-valued functions instead of the function banach spaces of the continuous real -valued functions .
For proof
The main part of the proof consists in the proof that the isometric isomorphism of the two function Banach spaces entails the homeomorphism of the two underlying compact Hausdorff spaces, because the proof of the reverse implication is simple. For this part of the evidence, however, in-depth aids from topology and functional analysis are required, in particular the following theorems :
- Kerin-Milman's theorem
- Banach-Alaoglu theorem
- Riesz's notation
- Lemma from Urysohn
It should be noted here that the lemma of Zorn (or an equivalent maximum principle of set theory ) is used to prove the Kerin-Milman theorem and to that extent also to prove the Banach-Stone theorem . A detailed presentation of the evidence can be found in Ronald Larsen's book Functional Analysis .
literature
Original work
- Jesús Araujo: The noncompact Banach-Stone theorem . In: Journal of Operator Theory . tape 55 , 2006, pp. 285-294 ( theta.ro [PDF; 136 kB ] MR2242851 ).
- MH Stone : Applications of the theory of boolean rings to General Topology . In: Trans. Amer. Math. Soc . tape 41 , 1937, pp. 375–481 ( ams.org [PDF; 12.0 MB ] MR1501905 ).
- M. Isabel Garrido, Jesús A. Jaramillo: Variations on the Banach-Stone theorem . In: Extracta Mathematicae . tape 17 , 2002, p. 351–383 ( eweb.unex.es [PDF; 282 kB ] MR1995413 ).
Monographs
- Stefan Banach: Théorie des opérations linéaires. Reprint of the first edition (Warszawa 1932) (= Chelsea scientific books . Volume 110 ). 2nd Edition. Chelsea Publishing, New York 1963, ISBN 0-8284-0110-1 .
- Bernard Beauzamy: Introduction to Banach Spaces and their Geometry. Unchanged reprint of the 1st edition from 1964 (= North-Holland Mathematics Studies . Volume 68 ). North-Holland Publishing Company, Amsterdam [u. a.] 1982, ISBN 0-444-86416-4 ( MR0670943 ).
- Gottfried Köthe : Topological linear spaces I (= The basic teachings of the mathematical sciences in individual representations . Volume 107 ). 2nd improved edition. Springer Verlag, Berlin [u. a.] 1966 ( MR0194863 ).
- Ronald Larsen: Functional Analysis. An Introduction (= Pure and Applied Mathematics . Volume 15 ). Marcel Dekker, New York 1973, ISBN 0-8247-6042-5 ( MR0461069 ).
- Dirk Werner : Functional Analysis (= The basic teachings of the mathematical sciences in individual representations . Volume 107 ). 6th corrected edition. Springer Verlag, Berlin [u. a.] 2007, ISBN 978-3-540-72536-7 .