On the structure of Lipschitz-free spaces

In this note we study the structure of Lipschitz-free Banach spaces. We show that every Lipschitz-free Banach space over an infinite metric space contains a complemented copy of \ell _1. This result has many consequences for the structure of Lipschitz-free Banach spaces. Moreover, we give an example of a countable compact metric space K such that \mathcal {F}(K) is not isomorphic to a subspace of L_1 and we show that whenever M is a subset of \mathbb{R}^n, then \mathcal {F}(M) is weakly sequentially complete; in particular, c_0 does not embed into \mathcal {F}(M).