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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Vector measure Banach spaces containing a complemented copy of $c_{0}$
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by A. Picón and C. Piñeiro PDF
Proc. Amer. Math. Soc. 132 (2004), 2893-2898 Request permission

Abstract:

Let $X$ a Banach space and $\Sigma$ a $\sigma$-algebra of subsets of a set $\Omega$. We say that a vector measure Banach space $(\mathcal {M} (\Sigma , X ) , \Vert \cdot \Vert _\mathcal {M })$ has the bounded Vitaly-Hahn-Sacks Property if it satisfies the following condition: Every vector measure $m : \Sigma \longrightarrow X$ , for which there exists a bounded sequence $(m_{n})$ in $\mathcal {M } (\Sigma , X )$ verifying $\displaystyle \lim _{n \to \infty } m_{n} ( A ) = m(A)$ for all $A \in \Sigma$, must belong to $\mathcal {M} (\Sigma , X)$. Among other results, we prove that, if $\mathcal {M}(\Sigma , X)$ is a vector measure Banach space with the bounded V-H-S Property and containing a complemented copy of $c_{0}$, then $X$ contains a copy of $c_{0}$.
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Additional Information
  • A. Picón
  • Affiliation: Departamento de Matemáticas, Facultad de Ciencias Experimentales, Campus universitario de El Carmen, Universidad de Huelva, 21071, Huelva, Spain
  • C. Piñeiro
  • Affiliation: Departamento de Matemáticas, Facultad de Ciencias Experimentales, Campus universitario de El Carmen, Universidad de Huelva, 21071, Huelva, Spain
  • Email: candido@uhu.es
  • Received by editor(s): May 14, 2002
  • Published electronically: May 21, 2004
  • Communicated by: Jonathan M. Borwein
  • © Copyright 2004 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 2893-2898
  • MSC (2000): Primary 46G10
  • DOI: https://doi.org/10.1090/S0002-9939-04-07518-5
  • MathSciNet review: 2063108