WEAK COMPACTNESS IN LIPSCHITZ-FREE SPACES OVER SUPERREFLEXIVE SPACES

We show that the Lipschitz-free space F(X) over a superreflexive Banach space X has the property that every weakly precompact subset of F(X) is relatively super weakly compact, showing that this space "behaves like L1" in this context. As consequences we show that F(X) enjoys the weak Banach-Saks property and that every subspace of F(X) with nontrivial type is superreflexive. It follows from our results that weakly compact subsets of F(X) are super weakly compact and hence have many strong properties. To prove the result, we use a modification of the proof of weak sequential completeness of F(X) by Kochanek and Perneck & aacute;and an appropriate version of compact reduction in the spirit of Aliaga, Nous, Petitjean and Proch & aacute;zka.