ON WEAKLY LINDELOF BANACH-SPACES

In this paper we define and investigate the properties of a proper class of Banach spaces each member of which is Lindelof in its weak topology; we call them weakly Lindelof determined (WLD) Banach spaces. The class of WLD Banach spaces extends the known class of WCD (weakly countably determined) Banach spaces and inherits some of its basic properties: (e.g., each WLD Banach space has a projectional resolution of identity, and it is also derived from a small weakly Lindelof subset, etc.). We also present several examples, in our attempt to clarify the concept of weakly countably determiness, such as: (i) a WLD Banach space which is dually strictly convexifiable, but not WCD; (ii) a WLD Banach with an unconditional basis, which is not weak Asplund, whose dual space is strictly convexifiable; (iii) a dual weakly K-analytic Banach space which is not a subspace of a weakly compactly generated Banach space. On the grounds of these examples, we answer questions and problems of Gruenhage, Larman and Phelps, and Talagrand.