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

### Antonio Jiménez-Vargas

Universidad de Almería, Spain

## 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 $τ_{bw_{∗}}$ 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 $γτ_{γ}$ on Lip$_{0}(X,F)$ such that the Lipschitz transpose mapping $f↦f_{t}$ is a topological isomorphism from Lip$_{0}(X,F),γτ_{γ})$ to (Lip$_{0}(X),τ_{bw_{∗}})ϵF$.

## Cite this article

Antonio Jiménez-Vargas, The approximation property for spaces of Lipschitz functions with the bounded weak* topology. Rev. Mat. Iberoam. 34 (2018), no. 2, pp. 637–654

DOI 10.4171/RMI/999