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

Abstract

We prove that for certain subsets $M\subseteq {\mathbb{R}}^N$, $N⩾1$, 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.

References in corpus (2)

• Lipschitz-free spaces over ultrametric spaces
• Free spaces over some proper metric spaces

Cited by in corpus (7)

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• Isomorphisms between spaces of Lipschitz functions
• Finitely additive measures and complementability of Lipschitz-free spaces
• Projecting Lipschitz functions onto spaces of polynomials
• Structure of the Lipschitz free $p$-spaces ${\mathcal{F}}_p({\mathbb{Z}}^d)$ and ${\mathcal{F}}_p({\mathbb{R}}^d)$ for $0<p\le 1$
• Tilings of the Hyperbolic Space and Lipschitz Functions
• On an identification of the Lipschitz-free spaces over subsets of ${\mathbb{R}}^n$