Subspaces of Lp, p>2, determined by partitions and weights

DOI10.4064/SM159-2-4zbMATH Open1057.46014arXivmath/0210228OpenAlexW1967790613MaRDI QIDQ4811110

Simei Tong, Dale Alspach

Publication date: 18 August 2004

Published in: Studia Mathematica (Search for Journal in Brave)

Abstract: Many of the known complemented subspaces of L_p have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well-known complemented subspaces of L_p. It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given by partitions and weights to be isomorphic to a subspace of L_p. Using this we define a space Y_n with norm given by partitions and weights with distance to any subspace of L_p growing with n. This allows us to construct an example of a Banach space with norm given by partitions and weights which is not isomorphic to a subspace of L_p.

Full work available at URL: https://arxiv.org/abs/math/0210228

zbMATH Keywords

envelope normcomplemented subspaces of \(L_p\)norms given by partitions and weights

Mathematics Subject Classification ID

Spaces of measurable functions ((L^p)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) (46E30) Geometry and structure of normed linear spaces (46B20)

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