Abstract
We prove that the Lipschitz-free space over a countable proper metric space is isometric to a dual space and has the metric approximation property. We also show that the Lipschitz-free space over a proper ultrametric space is isometric to the dual of a space which is isomorphic to \({c_0(\mathbb{N})}\) .
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References
Borel-Mathurin L.: Approximation properties and non-linear geometry of Banach space. Houst. J. Math. 38(4), 1135–1148 (2012)
Buneman P.: A note on the metric properties of trees. J. Comb. Theory Ser. B 17, 48–50 (1974)
Dalet, A.: Free spaces over countable compact metric spaces. Proc. Am. Math. Soc. (to appear)
Godard A.: Tree metrics and their Lipschitz-free spaces. Proc. Am. Math. Soc. 138(12), 4311–4320 (2010)
Godefroy G.: The use of norm attainment. Bull. Belg. Math. Soc. Simon Stevin 20(3), 417–423 (2013)
Godefroy G., Kalton N.J.: Lipschitz-free Banach spaces. Studia Math. 159(1), 121–141 (2003)
Godefroy G., Ozawa N.: Free Banach spaces and the approximation properties. Proc. Am. Math. Soc. 142(5), 1681–1687 (2014)
Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires. Mem. Am. Math. Soc. 16 (1955)
Hájek P., Pernecká E.: Schauder bases in Lipschitz-free spaces. J. Math. Anal. Appl. 416, 629–646 (2014)
Johnson W.B., Zippin M.: On subspaces of quotients of \({\left(\sum G_n\right)_{\ell_p}}\) and \({\left(\sum G_n\right)_{c_0}}\) . Isr. J. Math. 13, 311–316 (1972)
Kalton N.J.: Spaces of Lipschitz and Hölder functions and their applications. Collect. Math. 55(2), 171–217 (2004)
Kloeckner, B.R.: A geometric study of Wasserstein space: ultrametrics. Mathematika doi:10.1112/S0025579314000059
Lancien G., Pernecká E.: Approximation properties and Schauder decompositions in Lipschitz-free spaces. J. Funct. Anal. 264(10), 2323–2334 (2013)
Lewis D.R., Stegall C.: Banach spaces whose duals are isomorphic to \({l_{1}(\Gamma )}\) . J. Funct. Anal. 12, 177–187 (1973)
Lindenstrauss J., Tzafriri L.: Classical Banach Spaces I, Sequence Spaces. Springer, New York (1977)
Matoušek J.: Extension of Lipschitz mappings on metric trees. Comment. Math. Univ. Carol. 31(1), 99–104 (1990)
Naor A., Schechtman G.: Planar earthmover in not in L 1. SIAM J. Comput. 37(3), 804–826 (2007)
Pełczyński A.: Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basis. Studia Math. 40, 239–243 (1971)
Petunín Y.I., Plíčhko A.N.: Some properties of the set of functionals that attain a supremum on the unit sphere. Ukrain.Mat. Ž. 26, 102–106 (1974)
Weaver N.: Lipschitz algebras. World Scientific Publishing Co. Inc., River Edge (1999)
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The author was partially supported by PHC Barrande 26516YG.
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Dalet, A. Free Spaces Over Some Proper Metric Spaces. Mediterr. J. Math. 12, 973–986 (2015). https://doi.org/10.1007/s00009-014-0455-5
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DOI: https://doi.org/10.1007/s00009-014-0455-5