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Bases of rearrangement invariant spaces

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Abstract

We prove that if E is a rearrangement-invariant space, then a boundedly complete basis exists in E, if and only if one of the following conditions holds: 1) E is maximal and E ≠ L 1[0, 1]; 2) a certain (any) orthonormal system of functions from L ∞[0, 1], possessing the properties of the Schauder basis for the space of continuous on [0, 1] functions with the norm L ∞, represents a boundedly complete basis in E. As a corollary, we state the following assertion: Any (certain) orthonormal system of functions from L ∞[0, 1], possessing the properties of the Schauder basis for the space of continuous on [0, 1] functions with the norm L ∞, represents a spanning basis in a separable rearrangement-invariant space E, if and only if the adjoint space E* is separable. We prove that in any separable rearrangement-invariant space E the Haar system either forms an unconditional basis, or a strongly conditional one. The Haar system represents a strongly conditional basis in a separable rearrangement-invariant space, if and only if at least one of the Boyd indices of this space is trivial.

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Authors and Affiliations

  1. Universidad Autonoma de Madrid, Madrid, 28049, Spain

    K. S. Kazarian

  2. Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006, Russia

    E. M. Semenov & S. N. Uksusov

Authors
  1. K. S. Kazarian
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  2. E. M. Semenov
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  3. S. N. Uksusov
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Correspondence to K. S. Kazarian.

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Dedicated to the memory of the academician Petr Lavrent’evich Ul’yanov

Original Russian Text © K.S. Kazarian, E.M. Semenov, S.N. Uksusov, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 5, pp. 48–54.

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Kazarian, K.S., Semenov, E.M. & Uksusov, S.N. Bases of rearrangement invariant spaces. Russ Math. 52, 41–46 (2008). https://doi.org/10.3103/S1066369X08050058

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