Some remarks on the structure of Lipschitz-free spaces

TL;DR

This paper investigates the structure of Lipschitz-free spaces over various metric spaces, revealing their relationships with classical Banach spaces and their structural properties.

Contribution

It provides new structural results showing Lipschitz-free spaces contain complemented copies of classical spaces and are isomorphic to their squares or sums under certain conditions.

Findings

Contains complemented copy of ℓ₁(Γ)

Isomorphic to its square for nets in finite-dimensional spaces

Mutually isomorphic for spaces with Schauder basis

Abstract

We give several structural results concerning the Lipschitz-free spaces $\mathcal F(M)$, where $M$ is a metric space. We show that $\mathcal F(M)$ contains a complemented copy of $\ell_1(\Gamma)$, where $\Gamma=\text{dens}(M)$. If $\mathcal N$ is the net in a finite dimensional Banach space $X$, we show that $\mathcal F(\mathcal N)$ is isomorphic to its square. If $X$ contains a complemented copy of $\ell_p, c_0$ then $\mathcal F(\mathcal N)$ is isomorphic to its $\ell_1$-sum. Finally, we prove that for all $X\cong C(K)$ spaces $\mathcal F(\mathcal N)$ are mutually isomorphic spaces with a Schauder basis.