On spaces C^{b}(X) weakly K-analytic

Abstract

[EN] A subset Y of the dual closed unit ball B_{E*} of a Banach space E is called a Rainwater set for E if every bounded sequence of E that converges pointwise on Y converges weakly in E. In this paper, topological properties of Rainwater sets for the Banach space C^{b}(X) of the real-valued continuous and bounded functions defined on a completely regular space X equipped with the supremum-norm are studied. This applies to characterize the weak K-analyticity of C^{b}(X) in terms of certain Rainwater sets for C^{b}(X). Particularly, we show that C^{b}(X) is weakly K-analytic if and only if there exists a Rainwater set Y for C^{b}(X) such that (C^{b}(X),sigma_{Y}) is both K-analytic and angelic, where sigma_{Y} denotes the topology on C^{b}(X) of the pointwise convergence on Y. For the case when X is compact, one gets classic Talagrand¿s theorem. As an application we show that if X is a compact space and Y is a G_{delta}-dense subspace, then X is Talagrand compact, i.e., C^{p}(X)) is K-analytic, if and only if the space (C(X),sigma_{Y}) is K-analytic.