Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T22:06:54.606Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

AP.Alster, K. and Pol, R.. On function spaces of compact sets of Σ-products of the real line. Fund. Math., 107 (1980), 135–143.CrossRefGoogle Scholar
Ch.Choquet, G.. Lectures in Analysis (Benjamin, New York, 1969).Google Scholar
CK.Čoban, M. M. and Kenderov, P. S.. Generic Gateaux differentiability of convex functionals in C(T) and the topological properties of T. Preprint.Google Scholar
De.Debs, G.. Espaces k-analytiques et espaces de Baire de fonctions continues. Mathematika, 32 (1985), 218–228.CrossRefGoogle Scholar
En.Engelking, R.. General Topology (PWN, Warszawa, 1977).Google Scholar
Fr.Frolik, Z.. Baire spaces and some generalizations of complete metric spaces. Czech. Math. Jour., 20 (1970), 406–467.Google Scholar
Gr.Gruenhage, G.. A note on Gul'ko compact spaces. To appear.Google Scholar
Ja.James, R. C.. A separable somewhat reflexive Banach space with nonseparable dual. Bull Amer. Math. Soc., 80 (1974), 738–743.CrossRefGoogle Scholar
Ha.Haydon, R.. Some more characterizations of Banach spaces containing l 1. Proc. Camb. Phil. Soc., 80 (1976), 269–276.CrossRefGoogle Scholar
LS.Lindenstrauss, J. and Stegall, C.. Examples of separable spaces which do not contain l 1, and whose duals are nonseparable. Studia Math., 54 (1975), 81–105.CrossRefGoogle Scholar
Ne.Negrepontis, S.. Banach Spaces and Topology, in Handbook of Set Theoretic Topology (North Holland, Amsterdam, 1984).Google Scholar
P1.Pol, R.. On Pointwise and Weak Topology in Function Spaces (University of Warsaw, 1984).Google Scholar
P2.Pol, R.. Concerning function spaces on separable compact spaces. Bull. Acad. Pol. Sci., (1977), 993–998.Google Scholar
Ro.Rosenthal, H.. Pointwise compact sets of the first Baire class. Amer. Jour. Math., 99 (1977), 362–378.CrossRefGoogle Scholar
S1.Stegall, C.. A class of topological spaces and differentiation of functions in Banach spaces. Vorlesungen aus dem Fachbereich Mathematik der Universität Essen, 1983.Google Scholar
S2.Stegall, C.. Gateaux differentiation of functions of a certain class of Banach spaces, in Functional Analysis: Surveys and Recent Results III (North-Holland, Amsterdam, 1984).Google Scholar
S3.Stegall, C.. More Gateaux differentiability spaces, in Banach Spaces, Proceedings, Missouri 1984, Springer Lecture Notes No. 1166 (Berlin, 1985).Google Scholar
S4.Stegall, C.. Applications of Descriptive Topology in Functional Analysis (Universität Linz, 1985).Google Scholar
S5.Stegall, C.. The Radon-Nikodym property in conjugate Banach spaces. Trans. Amer. Math. Soc., 206 (1975), 213–223.CrossRefGoogle Scholar
To.Todorčević, S.. Trees and linearly ordered sets in Handbook of General Topology (North Holland, Amsterdam, 1984).Google Scholar