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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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Ja. R. C. James . A separable somewhat reflexive Banach space with nonseparable dual. Bull Amer. Math. Soc., 80 (1974), 738743.

Ne. S. Negrepontis . Banach Spaces and Topology, in Handbook of Set Theoretic Topology (North Holland, Amsterdam, 1984).

Ro. H. Rosenthal . Pointwise compact sets of the first Baire class. Amer. Jour. Math., 99 (1977), 362378.

S5. C. Stegall . The Radon-Nikodym property in conjugate Banach spaces. Trans. Amer. Math. Soc., 206 (1975), 213223.

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  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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