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Remarks on the set of norm-attaining functionals and differentiability

Volume 241 / 2018

Antonio J. Guirao, Vicente Montesinos, Vaclav Zizler Studia Mathematica 241 (2018), 71-86 MSC: Primary 46B20; Secondary 46B03, 46B26. DOI: 10.4064/sm8768-6-2017 Published online: 6 November 2017

Abstract

We use the smooth variational principle and a standard renorming to give a short direct proof of the classical Bishop–Phelps–Bollobás theorem on the density of norm-attaining functionals for weakly compactly generated Banach spaces. Then we show that a slight adjustment of a known Preiss–Zajíček differentiability argument provides a simple, useful characterization of individual norms on separable Banach spaces admitting residual sets of norm-attaining functionals in terms of Fréchet differentiability of their dual norms.

Authors

  • Antonio J. GuiraoInstituto de Matemática Pura y Aplicada

    Universitat Politècnica de València

    Camino de Vera, s/n

    46022 València, Spain
    e-mail
  • Vicente MontesinosInstituto de Matemática Pura y Aplicada

    Universitat Politècnica de València

    Camino de Vera, s/n

    46022 València, Spain
    e-mail
  • Vaclav ZizlerDepartment of Mathematics
    University of Alberta
    Central Academic Building
    Edmonton, AB, T6G 2G1, Canada
    e-mail

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