Authors
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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 and be Banach spaces. Suppose that is Asplund. Let be a bounded set of operators from to with the following property: a bounded sequence in is weakly null if, for each , the sequence is weakly null. Let be a sequence in such that: (a) for each , the set is relatively norm compact; (b) for each sequence in , the series is weakly unconditionally Cauchy. We prove that if is Dunford–Pettis and , then the series 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.
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