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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


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.


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

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