This article is cited in 5 papers

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

S. V. Astashkin^a, E. M. Semenov<sup>b

a</sup> Samara National Research University
^b Voronezh State University

Abstract: We study the family of rearrangement invariant spaces $E$ containing subspaces on which the $E$-norm is equivalent to the $L_1$-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 $E$.

Keywords: rearrangement invariant space, Marcinkiewicz space, Köthe dual space, subspace.

UDC: 517.982.27

Received: 25.02.2019

DOI: 10.4213/mzm12365

 English version: Mathematical Notes, 2020, 107:1, 10–19

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