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Compact reduction in Lipschitz-free spaces

Volume 260 / 2021

Ramón J. Aliaga, Camille Noûs, Colin Petitjean, Antonín Procházka Studia Mathematica 260 (2021), 341-359 MSC: Primary 46B20; Secondary 54E50. DOI: 10.4064/sm200925-18-1 Published online: 19 April 2021

Abstract

We prove a general principle satisfied by weakly precompact sets of Lips\-chitz-free spaces. By this principle, certain infinite-dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

Authors

  • Ramón J. AliagaUniversitat Politècnica de València
    Camino de Vera s/n
    46022 València, Spain
    e-mail
  • Camille NoûsLaboratoire Cogitamus
    e-mail
  • Colin PetitjeanLAMA, Univ Gustave Eiffel
    UPEM, Univ Paris Est Créteil, CNRS
    F-77447 Marne-la-Vallée, France
    e-mail
  • Antonín ProcházkaLaboratoire de Mathématiques
    de Besançon UMR 6623
    Université Bourgogne Franche-Comté, CNRS
    F-25000, Besançon, France
    e-mail

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