A renorming characterization of Banach spaces containing ℓ1(κ)

  1. Martínez-Cervantes, Gonzalo 2
  2. Rueda Zoca, Abraham 3
  3. Avilés López, Antonio
  1. 1 Universidad de Murcia. Departamento de Matemáticas
  2. 2 Universidad de Alicante. Departamento de Matemáticas
  3. 3 Universidad de Granada. Departamento de Análisis Matemático
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Revista:
Publicacions matematiques

ISSN: 0214-1493

Año de publicación: 2023

Volumen: 67

Número: 0

Páginas: 601-609

Tipo: Artículo

DOI: 10.5565/PUBLMAT6722305 DIALNET GOOGLE SCHOLAR lock_openDDD editor

Otras publicaciones en: Publicacions matematiques

Resumen

A result of G. Godefroy asserts that a Banach space X contains an isomorphic copy of `1 if and only if there is an equivalent norm ||| · ||| such that, for every finite-dimensional subspace Y of X and every ε > 0, there exists x ∈ SX so that |||y+rx||| ≥ (1−ε)(|||y|||+|r|) for every y ∈ Y and every r ∈ R. In this paper we generalise this result to larger cardinals, showing that if κ is an uncountable cardinal, then a Banach space X contains a copy of `1(κ) if and only if there is an equivalent norm ||| · ||| on X such that for every subspace Y of X with dens(Y ) < κ there exists a norm-one vector x so that |||y + rx||| = |||y||| + |r| whenever y ∈ Y and r ∈ R. This result answers a question posed by S. Ciaci, J. Langemets, and A. Lissitsin, where the authors wonder whether the above statement holds for infinite successor cardinals. We also show that, in the countable case, the result of Godefroy cannot be improved to take ε = 0.

Referencias bibliográficas

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