Applications of the quantification of super weak compactness

We introduce a measure of super weak noncompactness $\mathrm{\Gamma }$ defined for bounded linear operators and subsets in Banach spaces that allows to state and prove a characterization of the Banach spaces which are subspaces of a Hilbert generated space. The use of super weak compactness and $\mathrm{\Gamma }$ casts light on the structure of these Banach spaces and complements the work of Argyros, Fabian, Farmaki, Godefroy, H'ajek, Montesinos,\linebreak 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.