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

  • Antonio Jiménez-Vargas

    Universidad de Almería, Spain
The approximation property for spaces of Lipschitz functions with the bounded weak* topology cover
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Abstract

Let XX be a pointed metric space and let Lip0(X)_0(X) be the space of all scalar-valued Lipschitz functions on XX which vanish at the base point. We prove that Lip0(X)_0(X) with the bounded weak* topology τbw\tau_{bw^*} has the approximation property if and only if the Lipschitz-free Banach space F(X)\mathcal F(X) has the approximation property if and only if, for each Banach space FF, each Lipschitz operator from XX into FF can be approximated by Lipschitz finite-rank operators within the unique locally convex topology γτγ\gamma\tau_\gamma on Lip0(X,F)_0(X,F) such that the Lipschitz transpose mapping fftf\mapsto f^t is a topological isomorphism from Lip0(X,F),γτγ)_0(X,F),\gamma\tau_\gamma) to (Lip0(X),τbw)ϵF_0(X),\tau_{bw^*})\epsilon F.

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