On the structure of Lipschitz-free spaces

Cúth, Marek; Doucha, Michal; Wojtaszczyk, Przemysław

doi:10.1090/proc/13019

On the structure of Lipschitz-free spaces
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by Marek Cúth, Michal Doucha and Przemysław Wojtaszczyk

Proc. Amer. Math. Soc. 144 (2016), 3833-3846

DOI: https://doi.org/10.1090/proc/13019

Published electronically: February 17, 2016

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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 $\mathcal {F}(K)$ is not isomorphic to a subspace of $L_1$ and we show that whenever $M$ is a subset of $\mathbb {R}^n$, then $\mathcal {F}(M)$ is weakly sequentially complete; in particular, $c_0$ does not embed into $\mathcal {F}(M)$.

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