Bulletin of the Belgian Mathematical Society - Simon Stevin

Some remarks on the structure of Lipschitz-free spaces

Peter Hájek and Matěj Novotný

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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 a 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 $\f(\mathcal N)$ is isomorphic to its$\ell_1$-sum. Finally, we prove that for all $X\cong C(K)$ spaces, where $K$ is a metrizable compact, $\f(\mathcal N)$ are mutually isomorphic spaces with a Schauder basis.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 24, Number 2 (2017), 283-304.

Dates
First available in Project Euclid: 23 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1503453711

Digital Object Identifier
doi:10.36045/bbms/1503453711

Mathematical Reviews number (MathSciNet)
MR3694004

Zentralblatt MATH identifier
06850672

Subjects
Primary: 46B03: Isomorphic theory (including renorming) of Banach spaces 46B10: Duality and reflexivity [See also 46A25]

Citation

Hájek, Peter; Novotný, Matěj. Some remarks on the structure of Lipschitz-free spaces. Bull. Belg. Math. Soc. Simon Stevin 24 (2017), no. 2, 283--304. doi:10.36045/bbms/1503453711. https://projecteuclid.org/euclid.bbms/1503453711


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