We show that, given a Banach space $X$, the Lipschitz-free space over $X$, denoted by $\mathcal{ F}(X)$, is isomorphic to $(\sum_{ n=1}^\infty \mathcal{ F}(X))_{ \ell_1}$. Some applications are presented, including a non-linear version of Pelczy\'ski's decomposition method for Lipschitz-free spaces and the identification up to isomorphism between $\mathcal{ F}(\mathbb{ R}^n)$ and the Lipschitz-free space over any compact metric space which is locally bi-Lipschitz embeddable into $\mathbb{ R}^n$ and which contains a subset that is Lipschitz equivalent to the unit ball of $\mathbb{ R}^n$. We also show that $\mathcal{ F}(M)$ is isomorphic to $\mathcal{ F}(c_0)$ for all separable metric spaces $M$ which are absolute Lipschitz retracts and contain a subset which is Lipschitz equivalent to the unit ball of $c_0$. This class contains all $C(K)$ spaces with $K$ infinite compact metric (Dutrieux and Ferenczi had already proved that $\mathcal{ F}(C(K))$ is isomorphic to $\mathcal{ F}(c_0)$ for those $K$ using a different method). Finally we study Lipschitz-free spaces over certain unions and quotients of metric spaces, extending a result by Godard.
Categories: math.FA