This interesting article is devoted to a natural property of subsets of Banach spaces which has recently attracted attention. A weakly closed subset \(A\) of a Banach space \(X\) is called super weakly compact (in short, SWC) if its ultrapowers are relatively weakly compact in \(X^\mathcal{U}\) for each free ultrafilter. As demonstrated in this article, this notion yields to a nice interplay between local and topological properties of Banach spaces, related with classical theorems of James and Bourgain. Moreover, natural examples illustrate the relevance of this class of sets.
Here is a sample of some striking results shown in this work. A Banach space \(X\) is generated by some SWC set \(A\) (that is, the linear span of \(A\) is dense in \(X\)) if and only if \(X\) has an equivalent strongly uniformly Gâteaux-smooth norm. If moreover \(A\) strongly generates \(X\), then there is an equivalent norm on \(X\) whose restriction to any reflexive subspace of \(X\) is uniformly smooth and uniformly convex (Theorem 2.5). This result applies for instance to \(L_1\) spaces, and it implies in particular that reflexive subspaces of such a space \(X\) are super-reflexive. A closed convex set \(C\) is SWC if and only if real-valued Lipschitz functions defined on \(C\) are uniform limits of differences of convex Lipschitz functions (Theorem 3.5), if and only if it is contained in a uniformly convex set (Proposition 4.2). The existence of a uniform sequence of metric graphs with rates of change in \(C\) is equivalent to the fact that \(C\) is not SWC, for various classes of graphs: binary trees, diamond graphs, Laakso graphs (Theorem 5.1). Finally, several examples are considered in the last section: in particular every weakly compact subset of \(L_1\) is SWC and the Banach-Saks property follows (Corollary 6.4). Moreover, SWC subsets of \(c_0\) are investigated, and the Sauer-Shelah lemma provides a necessary condition on such sets (Proposition 6.5).