Weakly compactly generated spaces and their relatives

Marián Fabian, Prague, Czech Academy of Sciences

Abstract. A Banach space is called WCG it contains a weakly compact linearly dense subset. For instance, reflexive spaces or C(K), with K Eberlein compact, are such. The WCG spaces form an important subclass of nonseparable Banach spaces, with many nice properties. Their relatives are subspaces of WCG spaces, Vašák (i.e. WCD) spaces, weakly Lindelöf determined spaces, Hilbert generated spaces, subspaces of Hilbert generated spaces, ... We shall characterize each of these classes in terms of the presence of a total set therein with some extra properties. Other characterizations can be expressed via an equivalent norm with some kind of differentiability. A sample result: X is a subspace of a WCG space if and only if it admits a linearly dense set Γ ⊂ X and subsets Γ_{n}^{ε} ⊂ Γ, ε > 0, n ∈ IN, such that ⋃_{n∈IN} Γ_{n}^{ε} = Γ and for every x^{∗} ∈ X^{∗}, every ε > 0, and every n ∈ IN the set {γ ∈ Γ_{n}^{ε}; x^{∗}(γ) > ε} is finite. As a consequence we get a functionalanalytic proof of a well known fact that a continuous image of a (uniform) Eberlein compact is such. Our technique also reproves Farmaki’s result on sitting (uniform) Eberlein compacta in Σ(Γ).