K-2 - Banach Spaces and Topology (II)

Abstract

This chapter discusses Banach spaces and topology and stresses the interplay between the properties of weak and norm topologies. If a Banach space X is weakly Lindelöf determined (WCG), or more generally, if (X, w) is K-analytic, then (X, w) is a Lindelöf space. An important class of weak-Lindelöf Banach spaces are the so called WLD Banach spaces, and they coincide with Banach spaces with weak*-Corson compact dual unit ball. They provide a framework where AmirñLindenstrauss constructions for WCG Banach spaces can be carried out with analogous consequences. A Banach space X is said to have the property (C) (of Corson) if each family of closed convex subsets with the countable intersection property has non-empty intersection. This is a more general notion than that of weak-Lindelöf space. Whereas, property (C) is stable under taking finite products; it is yet an open problem to decide if the product of a weak-Lindelöf Banach space by itself is again weak-Lindelöf. In a Banach space, the norm can be changed to be equivalent without affecting the norm, weak, and weak* topologies.