Some remarks on the structure of Lipschitz-free spaces

Abstract

We give several structural results concerning the Lipschitz-free spaces $\mathcal{F}(M)$, where $M$ is a metric space. We show that $\mathcal{F}(M)$ contains a complemented copy of ${\ell }_1(\Gamma )$, where $\Gamma =\text{dens}(M)$. If $\mathcal{N}$ is the net in a finite dimensional Banach space $X$, we show that $\mathcal{F}(\mathcal{N})$ is isomorphic to its square. If $X$ contains a complemented copy of ${\ell }_p,c_0$ then $\mathcal{F}(\mathcal{N})$ is isomorphic to its ${\ell }_1$-sum. Finally, we prove that for all $X\cong C(K)$ spaces $\mathcal{F}(\mathcal{N})$ are mutually isomorphic spaces with a Schauder basis.