The Number of Weakly Compact Sets Which Generate a Banach Space

We consider the cardinal invariant CG(X) of the minimal number of weakly compact subsets which generate a Banach space X. We study the behavior of this index when passing to subspaces, its relation with the Lindelöf number in the weak topology and other related questions. A Banach space is weakly compactly generated if there is a weakly compact subset which is linearly dense and weakly Lindelöf if it is a Lindelöf space in its weak topology. It was asked by Corson [10] which was the relation between these two concepts. The answer was that every weakly compactly generated space is weakly Lindelöf but the converse is not true, and in order to clarify what was in the middle the class of weakly K-analytic was introduced by Talagrand [18], who was together with Pol [15] the first to solve this problem. Here we shall analyze the question of Corson from a more general point of view: What is the relation between the number of weak compacta which are necessary to generate a Banach space and the Lindelöf number of the space in the weak topology? Again, an intermediate class analagous to that introduced by Talagrand plays a clarifying role in the theory. Thus, our starting point is the following (cf. Sections 1 and 2 for notation): Definition 1. Let X be a topological space. (1) The index of compact generation of X , CG(X), is defined as the least infinite cardinal κ such that there exists a family {Kλ : λ < κ} of compact subsets of X whose union is a dense subset of X . (2) The index of K-analyticity of X , lK(X), is the least infinite cardinal κ for which there exists a complete metric space M of weight κ and an usco M −→ 2 . (3) The Lindelöf number of X , l(X), is the least infinite cardinal κ such that any cover of X by open sets has a subcover with at most κ many sets. If X is a Banach space, all the indices will refer always to the weak topology of X . In this way the classes of weakly compactly generated, weakly K-analytic and weakly Lindelöf Banach spaces equal the classes of spaces X such that CG(X) = ω, lK(X) = ω and l(X) = ω respectively. Similar indices to lK(X) can be defined if instead of complete metric spaces of a given weight we use other classes of topological spaces. These kind of indices have been studied in [13], cf. [9], such as the index of K-determinacy lΣ(X) (taking in (2) arbitrary metric spaces of weight κ instead of complete metric spaces) and the Nagami index Nag(X) (taking 2000 Mathematics Subject Classification. 46B26. This research was partially supported by the grant BFM2002-01719 of MCyT (Spain) and a FPU grant of MEC (Spain). 1