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On a Property of Rearrangement Invariant Spaces whose Second Köthe Dual is Nonseparable

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Abstract

We study the family of rearrangement invariant spaces E containing subspaces on which the E-norm is equivalent to the L1-norm and a certain geometric characteristic related to the Kadec–Pełcziński alternative is extremal. We prove that, after passing to an equivalent norm, any space with nonseparable second Köthe dual belongs to this family. In the course of the proof, we show that every nonseparable rearrangement invariant space E can be equipped with an equivalent norm with respect to which E contains a nonzero function orthogonal to the separable part of E.

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Funding

The first author’s work was supported by the Ministry of Education and Science of the Russian Federation (project no. 1.470.2016/1.4) and also, in part, by the Russian Foundation for Basic Research under grant 18-01-00414-a.

The second author’s work was supported by the Russian Foundation for Basic Research under grants 17-01-00138-a and 18-01-00414-a.

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Authors and Affiliations

  1. Korolev Samara National Research University, Samara, 443086, Russia

    S. V. Astashkin

  2. Voronezh State University, Voronezh, 394018, Russia

    E. M. Semenov

Authors
  1. S. V. Astashkin
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  2. E. M. Semenov
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Corresponding authors

Correspondence to S. V. Astashkin or E. M. Semenov.

Additional information

Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 1, pp. 11–22.

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Astashkin, S.V., Semenov, E.M. On a Property of Rearrangement Invariant Spaces whose Second Köthe Dual is Nonseparable. Math Notes 107, 10–19 (2020). https://doi.org/10.1134/S0001434620010022

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  • DOI: https://doi.org/10.1134/S0001434620010022

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