Lipschitz-free spaces and Schur properties

TL;DR

This paper investigates $ ext{Lipschitz-free spaces}$, revealing conditions for embedding into $ ext{l}_1$-sums, and characterizes when these spaces exhibit Schur properties, especially over metric spaces from $p$-Banach spaces.

Contribution

It provides new embedding results for Lipschitz-free spaces and establishes criteria for Schur properties, including for spaces over $p$-Banach spaces with $p$ in $(0,1)$.

Findings

Lipschitz-free spaces can embed into $ ext{l}_1$-sums of finite-dimensional subspaces.

Conditions for Lipschitz-free spaces to have the Schur, 1-Schur, and 1-strong Schur properties are identified.

Analysis of these properties over metric spaces derived from $p$-Banach spaces for $p$ in $(0,1)$.

Abstract

In this paper we study $\ell_1$-like properties for some Lipschitz-free spaces. The main result states that, under some natural conditions, the Lipschitz-free space over a proper metric space linearly embeds into an $\ell_1$-sum of finite dimensional subspaces of itself. We also give a sufficient condition for a Lipschitz-free space to have the Schur property, the $1$-Schur property and the $1$-strong Schur property respectively. We finish by studying those properties on a new family of examples, namely the Lipschitz-free spaces over metric spaces originating from $p$-Banach spaces, for $p$ in $(0,1)$.