On the structure of Lipschitz-free spaces

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 $F(M)$.

The only change in the latest version is the grant information of the second named author. Previous version contained a false proof of Theorem 1. This is corrected now. We have also added some remarks and changed Question 1, because we observed that the answer to the previous question is negative due to a result of P. L. Kaufmann. The paper has been accepted in Proc. Amer. Math. Soc

Cited by in corpus (17)

• Supports in Lipschitz-free spaces and applications to extremal structure
• On strongly norm attaining Lipschitz maps
• Generalized transportation cost spaces
• On relations between transportation cost spaces and ${\ell }_1$
• Isometric representation of Lipschitz-free spaces over convex domains in finite-dimensional spaces
• Isometric embedding of ${\ell }_1$ into Lipschitz-free spaces and ${\ell }_{\infty }$ into their duals
• Isomorphisms between spaces of Lipschitz functions
• Embeddability of ${\ell }_p$ and bases in Lipschitz free $p$-spaces for $0<p\le 1$
• Finitely additive measures and complementability of Lipschitz-free spaces
• Lipschitz-free spaces over compact subsets of superreflexive spaces are weakly sequentially complete
• Projections in Lipschitz-free spaces induced by group actions
• Hyperbolic Metric Spaces and Stochastic Embeddings
• Quantitative Schur property and measures of weak non-compactness
• Tilings of the Hyperbolic Space and Lipschitz Functions
• Note on almost isometric ideals and local retracts in Banach and metric spaces
• Lipschitz spaces over non-porous sets
• On the weak${}^{\ast }$ separability of the space of Lipschitz functions