TOPOLOGIES WEAKER THAN THE WEAK TOPOLOGY OF A BANACH-SPACE

[摘要] In this paper we investigate some properties of the compact subsets of Banach spaces X endowed with topologies of the kind sigma(X, B) where B is a norming subset of the dual unit ball B(X). Assuming that B(X). is sequentially compact we prove that the Krein-Smulian theorem holds for norm bounded sigma(X, B)-compact subsets of X and we state that the convex sigma(X, B)-compact subsets of X have the weak Radon-Nikodym property. When B(X) is sequentially compact and X has either the separable complementation property or X is weakly Lindelof (for instance, when B(X) is Corson compact) we prove that the sigma(X, B)-compact subsets (resp. sigma(X, B)-compact convex subsets) of X are fragmented by the norm of X (resp. have the Radon-Nikodym property). So, if B(X) is a Corson compact then the compact subsets of the space X[sigma(X; B)] are Radon-Nikodym compact and thus sequentially compact. We apply the previous results to prove that if B(X) is sequentially compact and B is assumed to be a boundary of B(X), then the norm bounded sigma(X, B)-compact subsets of X are weakly compact, which partially answers a problem posed by G. Godefroy. We give, among others, applications to spaces of vector-valued Bochner integrable functions as well as to spaces of countably additive measures. (C) 1994 Academic Press, Inc.

[发布日期] 1994-02-15