Separable measures and the Dunford-Pettis property

Abstract

Let X be a complete regular space. We denote by Cb(X) the Banach space of all real-valued bounded continuous functions on X endowed with the supremumnorm.In this paper we give a characterisation of weakly compact operators defined from Cb(X) into a Banach space E which are β∞-continuous, where β∞ is a locally convex topology on Cb(X) introduced by Wheeler. We also prove that (Cb(X), β∞) has the strict Dunford-Pettis property and, if X is a σ-compact space, (Cb(X), β∞), has the Dunford-Pettis property.