On the structure of Lipschitz-free spaces

Abstract

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 ℓ 1 \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 K such that F ( K ) \mathcal {F}(K) is not isomorphic to a subspace of L 1 L_1 and we show that whenever M M is a subset of R n \mathbb {R}^n , then F ( M ) \mathcal {F}(M) is weakly sequentially complete; in particular, c 0 c_0 does not embed into F ( M ) \mathcal {F}(M) .