The number of weakly compact sets which generate a Banach space

Abstract

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. IntroductionA 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. Corson [10] asked what the relation was 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, together with Pol [15], was 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 analogous 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):

Fiber orders and compact spaces of uncountable weight

“…For instance, for a finite power of the ball of the nonseparable Hilbert space B(κ) we get the following picture: Theorem 1. 2. Let K = B(κ) n and x = (x 1 , .…”

“…Then P (K) is first countable (for example by [13,Proposition 7]) while M(K) is not first countable as K is not metrizable. Therefore M(K) ≈ P (K) 2 . But we still can ask: Is M(K) ≈ P (K) 2 for compact spaces considered in this paper (A(κ), σ n (κ) etc.)?…”

“…Thus, the situation is completely analogous to that of P ([0, ω 1 ] n ), and we have the following result: Now we get easily the remaing part of Theorem 1.6 and Theorem 1.8. We mention that, answering a question posed to us by R. Deville and G. Godefroy, the ideas of this section are used in [2] to show that there exist 2 κ many nonhomeomorphic weakly compact convex subsets in ℓ 2 (κ).…”

“…We first had a long proof by induction of Lemma 5. 2. After speaking about it with Richard Haydon, he indicated to us a more elegant and shorter proof using combinatorial optimization that we reproduce below.…”

Some topological cardinal inequalities for spacesCp(X)

“…In [7] and [16] another useful cardinal function has been introduced which assigns to each topological space X the cardinal number Σ(X) of K-determination of X defined as the smallest infinite cardinal number λ for which there exist a metric space M with w(M ) = λ and an usco map from M into 2 X covering X. Both concepts have been essentially used to study and generalize several results from C p -theory, see [3], [7], [17]. Note that the spaces X with Σ(X) = ℵ 0 are precisely the Lindelöf Σ-spaces (also called K-determined spaces).…”