The approximation property for spaces of Lipschitz functions with the bounded weak* topology

Abstract

Let $X$ be a pointed metric space and let Lip${}_0(X)$ be the space of all scalar-valued Lipschitz functions on $X$ which vanish at the base point. We prove that Lip${}_0(X)$ with the bounded weak* topology ${\tau }_{b{w}^{\ast }}$ has the approximation property if and only if the Lipschitz-free Banach space $\mathcal{F}(X)$ has the approximation property if and only if, for each Banach space $F$, each Lipschitz operator from $X$ into $F$ can be approximated by Lipschitz finite-rank operators within the unique locally convex topology $\gamma {\tau }_{\gamma }$ on Lip${}_0(X,F)$ such that the Lipschitz transpose mapping $f\mapsto f^t$ is a topological isomorphism from Lip${}_0(X,F),\gamma {\tau }_{\gamma })$ to (Lip${}_0(X),{\tau }_{b{w}^{\ast }})ϵF$.