Free spaces over countable compact metric spaces

Dalet, A.

doi:10.1090/S0002-9939-2015-12518-X

Free spaces over countable compact metric spaces
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Proc. Amer. Math. Soc. 143 (2015), 3537-3546 Request permission

Abstract:

We prove that the Lipschitz-free space over a countable compact metric space is isometric to a dual space and has the metric approximation property.

References

Laetitia Borel-Mathurin, Approximation properties and non-linear geometry of Banach spaces, Houston J. Math. 38 (2012), no. 4, 1135–1148. MR 3019026
Gilles Godefroy, The use of norm attainment, Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 3, 417–423. MR 3129049
G. Godefroy and N. J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (2003), no. 1, 121–141. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday. MR 2030906, DOI 10.4064/sm159-1-6
Gilles Godefroy and Narutaka Ozawa, Free Banach spaces and the approximation properties, Proc. Amer. Math. Soc. 142 (2014), no. 5, 1681–1687. MR 3168474, DOI 10.1090/S0002-9939-2014-11933-2
Gilles Godefroy and Pierre David Saphar, Three-space problems for the approximation properties, Proc. Amer. Math. Soc. 105 (1989), no. 1, 70–75. MR 930249, DOI 10.1090/S0002-9939-1989-0930249-6
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
Petr Hájek and Eva Pernecká, On Schauder bases in Lipschitz-free spaces, J. Math. Anal. Appl. 416 (2014), no. 2, 629–646. MR 3188728, DOI 10.1016/j.jmaa.2014.02.060
N. J. Kalton, Spaces of Lipschitz and Hölder functions and their applications, Collect. Math. 55 (2004), no. 2, 171–217. MR 2068975
G. Lancien and E. Pernecká, Approximation properties and Schauder decompositions in Lipschitz-free spaces, J. Funct. Anal. 264 (2013), no. 10, 2323–2334. MR 3035057, DOI 10.1016/j.jfa.2013.02.012
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
Assaf Naor and Gideon Schechtman, Planar earthmover is not in $L_1$, SIAM J. Comput. 37 (2007), no. 3, 804–826. MR 2341917, DOI 10.1137/05064206X
A. Pełczyński, Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basis, Studia Math. 40 (1971), 239–243. MR 308753, DOI 10.4064/sm-40-3-239-243
Ju. Ī. Petunīn and A. N. Plīčko, Some properties of the set of functionals that attain a supremum on the unit sphere, Ukrain. Mat. Ž. 26 (1974), 102–106, 143 (Russian). MR 0336299
Nik Weaver, Lipschitz algebras, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR 1832645, DOI 10.1142/4100