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Topological spaces with dense subspaces that are homeomorphic to complete metric spaces and the classification of C(K) Banach spaces

Part of: Linear function spaces and their duals

Published online by Cambridge University Press:  26 February 2010

Charles Stegall
Charles Stegall
Affiliation:
Institut für Mathematik, Johannes Kepler Universität, A-4040, Linz, Austria.
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Extract

In [S1] we introduced and in [S2, S3, S4] developed a class of topological spaces that is useful in the study of the classification of Banach spaces and Gateaux differentiation of functions defined in Banach spaces. The class C may be most succinctly defined in the following way: a Hausdorff space T is in C if any upper semicontinuous compact valued map (usco) that is minimal and defined on a Baire space B with values in T must be point valued on a dense Gδ subset of B. This definition conceals many interesting properties of the family C. See [S2] for a discussion of the various definitions. Our main result here is that if X is a Banach space such that the dual space X* in the weak* topology is in C and K is any weak* compact subset of X* then the extreme points of K contain a dense, necessarily Gδ, subset homeomorphic to a complete metric space. In [S4] we studied the class K of κ-analytic spaces in C. Here we shall show that many elements of K contain dense subsets homeomorphic to complete metric spaces. It is easy to see that C contains all metric spaces and it is proved in [S4] that analytic spaces are in K. We obtain a number of topological results that may be of independent interest. We close with a discussion of various examples that show the interaction of these ideas between functional analysis and topology

MSC classification

Secondary: 46E15: Banach spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Mathematika , Volume 34 , Issue 1 , June 1987 , pp. 101 - 107
Copyright
Copyright © University College London 1987

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