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On the structure of Lipschitz-free spaces


Authors: Marek Cúth, Michal Doucha and Przemysław Wojtaszczyk
Journal: Proc. Amer. Math. Soc. 144 (2016), 3833-3846
MSC (2010): Primary 46B03, 54E35
DOI: https://doi.org/10.1090/proc/13019
Published electronically: February 17, 2016
MathSciNet review: 3513542
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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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Additional Information

Marek Cúth
Affiliation: Instytut Matematyczny Polskiej Akademii Nauk, Śniadeckich 8, 00-656 Warszawa, Poland
Address at time of publication: Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83, 186 75 Prague 8, Czech Republic
Email: cuth@karlin.mff.cuni.cz

Michal Doucha
Affiliation: Instytut Matematyczny Polskiej Akademii Nauk, Śniadeckich 8, 00-656 Warszawa, Poland
Email: m.doucha@post.cz

Przemysław Wojtaszczyk
Affiliation: Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw, ul. Prosta 69, 02-838 Warszawa, Poland
Email: wojtaszczyk@icm.edu.pl

DOI: https://doi.org/10.1090/proc/13019
Keywords: Lipschitz-free space, isomorphically universal separable Banach space, embedding of $c_0$
Received by editor(s): May 27, 2015
Received by editor(s) in revised form: July 14, 2015, and October 28, 2015
Published electronically: February 17, 2016
Additional Notes: The first author was supported by the Warsaw Center of Mathematics and Computer Science (KNOW–MNSzW)
The second author was supported by IMPAN’s International Fellowship Programme and partially sponsored by PCOFUND-GA-2012-600415.
The third author was supported by Polish NCN grant UMO-2011/03/B/ST1/04902.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2016 American Mathematical Society