Lipschitz-free spaces and Schur properties

In this paper we study l(1)-like properties for some Lipschitz-free spaces. The main result states that, under some natural conditions, the Lipschitz-free space over a proper metric space linearly embeds into an l(1)-sum of finite dimensional subspaces of itself. We also give a sufficient condition for a Lipschitz-free space to have the Schur property, the 1-Schur property and the 1-strong Schur property respectively. We finish by studying those properties on a new family of examples, namely the Lipschitz-free spaces over metric spaces originating from p-Banach spaces, for p in (0,1). (C) 2017 Elsevier Inc. All rights reserved.