Lipschitz-free spaces are strongly unique preduals

Abstract

We prove that the Lipschitz-free space $\mathcal{F}(M)$ is the strongly unique isometric predual of $\mathrm{Lip}_0(M)$ for every metric space $M$, solving a longstanding open problem of Weaver. The result is proved first for length metric spaces by analyzing the behavior of Lipschitz functions on paths in $M$ and the topological properties of subspaces of functions vanishing on certain barrier-sets of $M$. Then, the result is extended to general metric spaces by embedding them into length spaces in such a way that the restriction operator for Lipschitz functions is weak$^*$ continuous with respect to their selected preduals. Finally, we also give a counterexample to the codimension-one inheritance assertion used in Weaver's original proof for bounded and geodesic metric spaces.