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

  • Charles Stegall (a1)

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

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AP.Alster K. and Pol R.. On function spaces of compact sets of Σ-products of the real line. Fund. Math., 107 (1980), 135143.
Ch.Choquet G.. Lectures in Analysis (Benjamin, New York, 1969).
CK.Čoban M. M. and Kenderov P. S.. Generic Gateaux differentiability of convex functionals in C(T) and the topological properties of T. Preprint.
De.Debs G.. Espaces k-analytiques et espaces de Baire de fonctions continues. Mathematika, 32 (1985), 218228.
En.Engelking R.. General Topology (PWN, Warszawa, 1977).
Fr.Frolik Z.. Baire spaces and some generalizations of complete metric spaces. Czech. Math. Jour., 20 (1970), 406467.
Gr.Gruenhage G.. A note on Gul'ko compact spaces. To appear.
Ja.James R. C.. A separable somewhat reflexive Banach space with nonseparable dual. Bull Amer. Math. Soc., 80 (1974), 738743.
Ha.Haydon R.. Some more characterizations of Banach spaces containing l 1. Proc. Camb. Phil. Soc., 80 (1976), 269276.
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), 81105.
Ne.Negrepontis S.. Banach Spaces and Topology, in Handbook of Set Theoretic Topology (North Holland, Amsterdam, 1984).
P1.Pol R.. On Pointwise and Weak Topology in Function Spaces (University of Warsaw, 1984).
P2.Pol R.. Concerning function spaces on separable compact spaces. Bull. Acad. Pol. Sci., (1977), 993998.
Ro.Rosenthal H.. Pointwise compact sets of the first Baire class. Amer. Jour. Math., 99 (1977), 362378.
S1.Stegall C.. A class of topological spaces and differentiation of functions in Banach spaces. Vorlesungen aus dem Fachbereich Mathematik der Universität Essen, 1983.
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).
S3.Stegall C.. More Gateaux differentiability spaces, in Banach Spaces, Proceedings, Missouri 1984, Springer Lecture Notes No. 1166 (Berlin, 1985).
S4.Stegall C.. Applications of Descriptive Topology in Functional Analysis (Universität Linz, 1985).
S5.Stegall C.. The Radon-Nikodym property in conjugate Banach spaces. Trans. Amer. Math. Soc., 206 (1975), 213223.
To.Todorčević S.. Trees and linearly ordered sets in Handbook of General Topology (North Holland, Amsterdam, 1984).
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  • ISSN: 0025-5793
  • EISSN: 2041-7942
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