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Approximation properties in terms of Lipschitz maps

Volume 268 / 2023

Mingu Jung, Ju Myung Kim Studia Mathematica 268 (2023), 345-359 MSC: Primary 46B28; Secondary 46B45, 47L20. DOI: 10.4064/sm220314-19-8 Published online: 10 October 2022

Abstract

We investigate some approximation properties of Banach spaces which are described in terms of Lipschitz maps. First, we present characterizations of the Lipschitz approximation property, and prove that a Banach space $X$ has the approximation property whenever the Lipschitz-free space over $X$ has this property. Furthermore, we obtain a Lipschitz version of Grothendieck’s characterization of the classical approximation property. Second, we introduce the Lipschitz weak $\lambda $-bounded approximation property and show that it implies the classical weak $\lambda $-bounded approximation property. Finally, several equivalent formulations of the Lipschitz weak $\lambda $-bounded approximation property are obtained.

Authors

  • Mingu JungSchool of Mathematics
    Korea Institute for Advanced Study
    02455 Seoul, Republic of Korea
    ORCID: 0000-0003-2240-2855
    e-mail
  • Ju Myung KimDepartment of Mathematics and Statistics
    Sejong University
    Seoul 05006, Republic of Korea
    e-mail

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