On the structure of Lipschitz-free spaces

TL;DR

This paper investigates the structure of Lipschitz-free Banach spaces, revealing that over infinite metric spaces they contain complemented copies of ℓ₁, and explores their properties and embeddings in various contexts.

Contribution

It demonstrates that Lipschitz-free Banach spaces over infinite metric spaces contain complemented ℓ₁, provides examples of non-isomorphic spaces, and analyzes their weak sequential completeness.

Findings

Lipschitz-free Banach spaces over infinite metric spaces contain complemented ℓ₁.

There exists a countable compact metric space whose Lipschitz-free space is not isomorphic to a subspace of L₁.

F(M) is weakly sequentially complete for subsets M of ℝⁿ, and c₀ does not embed into F(M).

Abstract

In this note we study the structure of Lipschitz-free Banach spaces. We show that every Lipschitz-free Banach space over an infinite metric space contains a complemented copy of $\ell_1$. This result has many consequences for the structure of Lipschitz-free Banach spaces. Moreover, we give an example of a countable compact metric space $K$ such that $F(K)$ is not isomorphic to a subspace of $L_1$ and we show that whenever $M$ is a subset of $R^n$, then $F(M)$ is weakly sequentially complete; in particular, $c_0$ does not embed into $F(M)$.