PRODUCTS OF LIPSCHITZ-FREE SPACES AND APPLICATIONS

Abstract. We show that, given a Banach space X, the Lipschitz-free space over X, denoted by F(X), is isomorphic to (∑∞n=1 F(X))`1. Some applications are presented, in-cluding a non-linear version of Pe lczyśki’s decomposition method for Lipschitz-free spaces and the identification up to isomorphism between F(Rn) and the Lipschitz-free space over any compact metric space which is locally bi-Lipschitz embeddable into Rn and which con-tains a subset that is Lipschitz equivalent to the unit ball of Rn. We also show that F(M) is isomorphic to F(c0) for all separable metric spaces M which are absolute Lipschitz re-tracts and contain a subset which is Lipschitz equivalent to the unit ball of c0. This class contains all C(K) spaces with K infinite compact metric (Dutrieux and Ferenczi had al-ready proved that F(C(K)) is isomorphic to F(c0) for those K using a different method). Finally we study Lipschitz-free spaces over certain unions and quotients of metric spaces, extending a result by Godard. 1