A note on summability in Banach spaces

Authors

  • José Rodríguez Universidad de Castilla-La Mancha, Dpto. de Matemáticas, E.T.S. de Ingenieros Industriales de Albacete

Keywords:

Absolutely convergent series, Dunford–Pettis operator, vector measure, Schauder basis

Abstract

Let Z and X be Banach spaces. Suppose that X is Asplund. Let M be a bounded set of operators from Z to X with the following property: a bounded sequence (zn)nN in Z is weakly null if, for each MM, the sequence (M(zn))nN is weakly null. Let (zn)nN be a sequence in Z such that: (a) for each nN, the set {M(zn):MM} is relatively norm compact; (b) for each sequence (Mn)nN in M, the series n=1Mn(zn) is weakly unconditionally Cauchy. We prove that if TM is Dunford–Pettis and infnNT(zn)zn1>0, then the series n=1T(zn) is absolutely convergent. As an application, we provide another proof of the fact that a countably additive vector measure taking values in an Asplund Banach space has finite variation whenever its integration operator is Dunford–Pettis.
Section
Articles

Published

2025-01-30

How to Cite

Rodríguez, J. (2025). A note on summability in Banach spaces. Annales Fennici Mathematici, 50(1), 49–58. https://doi.org/10.54330/afm.156613