The Dunford-Pettis property for Lipschitz-free spaces over $\mathbb{R}^n$

Abstract

We show that for every $n\in \mathbb{N}$, the Lipschitz-free space $\mathcal{F}(\mathbb{R}^n)$ has the Dunford-Pettis property. As a consequence, for every $n\in \mathbb{N}$ and every nonempty $M\subset \mathbb{R}^n$, the space $\mathcal{F}(M)$ has the Dunford-Pettis property. This implies that $\mathcal{F}(M)$ is not isomorphic to $\mathcal{F}(X)$ for any separable Banach space $X$ that fails the Dunford-Pettis property. In particular, this applies to $X=\ell_2$, and more generally to every infinite-dimensional separable reflexive space. We also deduce that if $1<p<\infty$, then $\mathcal{F}(\ell_p)$ is not isomorphic to $\left(\bigoplus\limits_{n=1}^\infty \mathcal{F}(\ell_p^n) \right)_{\ell_1}$.