Bulletin of the Belgian Mathematical Society - Simon Stevin

Approximation and Schur properties for Lipschitz free spaces over compact metric spaces

P. Hájek, G. Lancien, and E. Pernecká

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Abstract

We prove that for any separable Banach space $X$, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to $X$. As a consequence we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails the approximation property and we prove that there exists an uncountable family of topologically equivalent distances on the Cantor space whose free spaces are pairwise non isomorphic. We also prove that the free space over a countable compact metric space has the Schur property. These results answer questions by G. Godefroy.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 23, Number 1 (2016), 63-72.

Dates
First available in Project Euclid: 9 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1457560854

Mathematical Reviews number (MathSciNet)
MR3471979

Zentralblatt MATH identifier
1353.46013

Subjects
Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B80: Nonlinear classification of Banach spaces; nonlinear quotients

Keywords
Lipschitz free spaces approximation property Schur property Cantor space

Citation

Hájek, P.; Lancien, G.; Pernecká, E. Approximation and Schur properties for Lipschitz free spaces over compact metric spaces. Bull. Belg. Math. Soc. Simon Stevin 23 (2016), no. 1, 63--72. https://projecteuclid.org/euclid.bbms/1457560854


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