Subspaces of Hilbert-generated Banach spaces and the quantification of super weak compactness

Abstract

We introduce a measure of super weak noncompactness Γ defined for bounded subsets and bounded linear operators in Banach spaces that allows to state and prove a charac-terization of the Banach spaces which are subspaces of a Hilbert-generated space. The use of super weak compactness and Γcasts light on the structure of these Banach spaces and complements the work of Argyros, Fabian, Farmaki, Godefroy, Hájek, Montesinos, Troyanski and Zizler on this subject. A particular kind of relatively super weakly compact sets, namely uniformly weakly null sets, plays an important role and exhibits connections with Banach-Saks type properties.