Inner characterizations of weakly compactly generated Banach spaces and their relatives

Abstract

We give characterizations of weakly compactly generated spaces, their subspaces, Vašák spaces, weakly Lindelöf determined spaces as well as several other classes of Banach spaces related to uniform Gâteaux smoothness, in terms of the presence of a total subset of the space with some additional properties. In addition, we describe geometrically, when possible, these classes by means of suitable smoothness or rotundity of the norm. As a consequence, we get new, functional analytic proofs of several theorems on (uniform) Eberlein, Gul'ko and Talagrand compacta.

A note on weakly Lindelöf determined Banach spaces

“…It is known that a WLD Banach space has such a PG (see, for example, [2, Proposition 8.3.1]). That a WLD is characterized by the existence of an M-basis (or just a linearly dense subset) that countably supports X * is also known (see, for example, [3] or [4,Theorem 12.50]).…”

Weak compactness and σ-Asplund generated Banach spaces

“…Then [9,Theorem 2] shows that the ball (B C(K) * , w * ) is an Eberlein compact space, and hence, a fortiori, so is K.…”

“…Definitions and notation. In [9], the results on quantitative versions of the differentiability of norms and of weak compactness were used to give characterizations of several subclasses of weakly Lindelöf determined spaces.…”

“…Note that if X is WLD, then it contains a linearly dense set Γ ⊂ B X which countably supports all elements of X * , that is, for every x * ∈ X * the set {γ ∈ Γ ; γ, x * = 0} is at most countable [9,Theorem 5]. In order to prove (o), by [9,Theorem 2], it suffices to show the following Claim 1.…”

“…Note that if X is WLD, then it contains a linearly dense set Γ ⊂ B X which countably supports all elements of X * , that is, for every x * ∈ X * the set {γ ∈ Γ ; γ, x * = 0} is at most countable [9,Theorem 5]. In order to prove (o), by [9,Theorem 2], it suffices to show the following Claim 1. There exists a linearly dense set Γ ⊂ B X such that for every ε > 0 there are subsets Γ ε i ⊂ Γ, i ∈ N, satisfying Γ = ∞ n=1 Γ ε n , and such that for all n ∈ N and x * ∈ X * the set {γ ∈ Γ ε n ; γ, x * > ε} is finite.…”