The Metric Approximation Property and Lipschitz-Free Spaces over Subsets of $\mathbb{R}^N$

Eva Pernecká; Richard J. Smith

doi:10.1016/j.jat.2015.06.003

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arXiv · 1501.07036

The Metric Approximation Property and Lipschitz-Free Spaces over Subsets of $\mathbb{R}^N$

Eva Pernecká·Richard J. Smith

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Abstract

We prove that for certain subsets $M \subseteq \mathbb{R}^N$, $N \geqslant 1$, the Lipschitz-free space $\mathcal{F}(M)$ has the metric approximation property (MAP), with respect to any norm on $\mathbb{R}^N$. In particular, $\mathcal{F}(M)$ has the MAP whenever $M$ is a finite-dimensional compact convex set. This should be compared with a recent result of Godefroy and Ozawa, who showed that there exists a compact convex subset $M$ of a separable Banach space, for which $\mathcal{F}(M)$ fails the approximation property.

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BibTeX RIS

Eva Pernecká, Richard J. Smith. 2015-01-28. The Metric Approximation Property and Lipschitz-Free Spaces over Subsets of $\mathbb{R}^N$. https://doi.org/10.1016/j.jat.2015.06.003

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