On spaces weakly ‐analytic

Abstract

AbstractA subset Y of the dual closed unit ball 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 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 in terms of certain Rainwater sets for . Particularly, we show that is weakly K‐analytic if and only if there exists a Rainwater set Y for such that is both K‐analytic and angelic, where denotes the topology on 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 ‐dense subspace, then X is Talagrand compact, i.e., is K‐analytic, if and only if the space is K‐analytic.