The compact weak topology on a Banach space

Throughout [this paper], E and F will denote Banach spaces. The bounded weak topology on a Banach space E, noted bw(E) or simply bw, is defined as the finest topology that agrees with the weak topology on bounded sets. It is proved in [3] that bw(E) is a locally convex topology if and only if E is reflexive. In this paper we introduce the compact weak topology on a Banach space E, noted kw(E) or simply kw, as the finest topology that agrees with the weak topology on weakly compact subsets. Equivalently, kw is the finest topology having the same convergent sequences as the weak topology. This topology appears in a natural manner in the study of a certain class of continuous mappings. We prove that kw(E) is a locally convex topology if and only if the space E is reflexive or has the Schur property. We denote ckw the finest locally convex topology contained in kw, and derive characterizations of Banach spaces not containing l1, and of other classes of Banach spaces, in terms of these topologies. It is also shown that ckw(E) is the topology of uniform convergence on (L)-sets of the dual space E*. As a consequence, we characterize Banach spaces with the reciprocal Dunford-Pettis property