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Proceedings of the American Mathematical Society

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



Banach spaces in which every compact lies inside the range of a vector measure

Authors: C. Piñeiro and L. Rodríguez-Piazza
Journal: Proc. Amer. Math. Soc. 114 (1992), 505-517
MSC: Primary 46B20; Secondary 46G10
MathSciNet review: 1086342
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Abstract: We prove that the compact subsets of a Banach space $ X$ lie inside ranges of $ X$-valued measures if and only if $ {X^*}$ can be embedded in an $ {L^1}$ space. In these spaces we prove that every compact is, in fact, a subset of a compact range. We also prove that if every compact of $ X$ is a subset of the range of an $ X$-valued measure of bounded variation, then $ X$ is finite dimensional. Thus we answer a question by R. Anantharaman and J. Diestel.

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Keywords: Vector measures, range, Banach spaces, compact sets, subspaces of $ {L^1}$
Article copyright: © Copyright 1992 American Mathematical Society