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

We introduce a measure of super weak noncompactness Gamma 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 Gamma casts light on the structure of these Banach spaces and complements the work of Argyros, Fabian, Farmaki, Gode-froy, Hajek, 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.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).