Lipschitz-free spaces, ultraproducts, and finite representability of metric spaces

TL;DR

This paper explores the properties of ultrapowers of metric spaces and their Lipschitz-free spaces, establishing finite representability results and characterizations that connect metric space embeddings with Banach space properties.

Contribution

It introduces new characterizations of finitely Lipschitz representable metric spaces via ultrapowers and links these to Lipschitz-free space properties, advancing understanding of ultraproduct stability.

Findings

Lipschitz-free space of ultrapower is finitely representable in the original space

Finitely Lipschitz representable metric spaces embed into ultrapowers of Banach spaces

Results apply to cotype and stability of Lipschitz-free spaces under ultraproducts

Abstract

We study several properties and applications of the ultrapower $M_{\mathcal U}$ of a metric space $M$. We prove that the Lipschitz-free space $\mathcal F(M_{\mathcal U})$ is finitely representable in $\mathcal F(M)$. We also characterize the metric spaces that are finitely Lipschitz representable in a Banach space as those that biLipschitz embed into an ultrapower of the Banach space. Thanks to this link, we obtain that if $M$ is finitely Lipschitz representable in a Banach space $X$, then $\mathcal F(M)$ is finitely representable in $\mathcal F(X)$. We apply these results to the study of cotype in Lipschitz-free spaces and the stability of Lipschitz-free spaces and spaces of Lipschitz functions under ultraproducts.