# Revista Matemática Iberoamericana

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**Volume 34, Issue 2, 2018, pp. 637–654**

**DOI: 10.4171/RMI/999**

Published online: 2018-05-28

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

Antonio Jiménez-Vargas^{[1]}(1) Universidad de Almería, Spain

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 $\tau_{bw^*}$ 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$_0(X),\tau_{bw^*})\epsilon F$.

*Keywords: *Lipschitz spaces, approximation property, tensor product, epsilon product

Jiménez-Vargas Antonio: The approximation property for spaces of Lipschitz functions with the bounded weak* topology. *Rev. Mat. Iberoam.* 34 (2018), 637-654. doi: 10.4171/RMI/999