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Geometry of Banach spaces with property β

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Abstract

We prove that every Banach space can be isometrically and 1-complementably embedded into a Banach space which satisfies property β and has the same character of density. Then we show that, nevertheless, property β satisfies a hereditary property. We study strong subdifferentiability of norms with property β to characterize separable polyhedral Banach spaces as those admitting a strongly subdifferentiable β norm. In general, a Banach space with such a norm is polyhedral. Finally, we provide examples of non-reflexive spaces whose usual norm fails property β and yet it can be approximated by norms with this property, namely (L 1[0,1], ‖·‖1) and (C(K), ‖·‖∗) whereK is a separable Hausdorff compact space

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Authors and Affiliations

  1. Departamento de Análisis, Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040, Madrid, Spain

    A. S. Granero & M. Jiménez Sevilla

  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma, 28049, Madrid, Spain

    J. P. Moreno

Authors
  1. A. S. Granero
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  2. M. Jiménez Sevilla
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  3. J. P. Moreno
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Corresponding author

Correspondence to A. S. Granero.

Additional information

To the memory of A. Plans

Supported in part by DGICYT grant PB 94-0243 and DGICYT PB 96-0607.

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Granero, A.S., Sevilla, M.J. & Moreno, J.P. Geometry of Banach spaces with property β. Isr. J. Math. 111, 263–273 (1999). https://doi.org/10.1007/BF02810687

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  • DOI: https://doi.org/10.1007/BF02810687

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