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 basepoint. We prove that Lip_0(X) with the bounded weak* topology τ_{bw∗} has the approximation property if and only if the Lipschitz-free Banach space 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→f^t is a topological isomorphism from (Lip_(X,F),\gamma\tau_{\gamma}) to (Lip_0(X), τ_{bw∗})\epsilon F .