A Grothendieck-Type Compactness Principle for Super Weakly Compact Sets

The classical Grothendieck compactness principle states that every norm-compact subset of a Banach space lies in the closed convex hull of a norm null sequence. Replacing the norm topology by the weak topology yields a characterization of Banach spaces with the Schur property. In the present paper, we establish an extension of this principle to the framework of super weakly compact sets. Specifically, we prove that for a Banach space X with the weak Banach–Saks property, every super weakly compact subset of X is contained in the closed convex hull of a uniformly weakly null sequence if and only if X has the Schur property. In addition, we demonstrate that every Banach space failing the Schur property contains a weakly null sequence which is not uniformly weakly null.