ON LARGE l 1 -SUMS OF LIPSCHITZ-FREE SPACES AND APPLICATIONS

Abstract

We prove that the Lipschitz-free space over a Banach space X of density κ, denoted by F (X), is linearly isomorphic to the l 1 -sum of κ copies of F (X) . This provides an extension of a previous result from Kaufmann in the context of non-separable Banach spaces. Further, we obtain a complete classification of the spaces of real-valued Lipschitz functions that vanish at 0 over a L p -space. More precisely, we establish that, for every 1 ≤ p ≤ ∞, if X is a L p -space of density κ, then Lip 0 (X) is either isomorphic to Lip 0 (l p (κ)) if p < ∞, or Lip 0 (c 0 (κ)) if p = ∞.