On Weakly Lindelöf Banach Spaces

Abstract

The purpose of this paper is to show how the method, developed by M. Valdivia, of constructing projections in a weakly compactly generated Banach space X can be used to give a direct proof of the Lindelof property for the weak topology in X. Since this method extends to large classes of spaces we prove a Lindelof property for Banach spaces with a Valdivia compact weak* dual unit ball. Our result includes previous ones of K. Alster, R. Pol and S. P. Gul'ko for the space C(K) with the topology of pointwise convergence, where K is a Corson compact space. If X is a Banach space then the dual X* has the Radon-Nikodym property if and only if it is Lindelof in the topology of uniform convergence on the separable bounded subsets of X. Finally, if a dual space X* is weakly Lindelof then the bidual unit ball must be a Corson compact space in the weak* topology, so the product X* ×X* is weakly Lindelof, too. This result gives a positive answer to a problem of Corson in the case of dual spaces.