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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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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A geometric characterization of Fréchet spaces with the Radon-Nikodým property
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by G. Y. H. Chi PDF
Proc. Amer. Math. Soc. 48 (1975), 371-380 Request permission

Abstract:

Let $F$ be a locally convex Fréchet space. $F$ is said to have the Radon-Nikodym property if for every positive finite measure space $(\mathbf {\Omega ,\Sigma },\mu )$, and every $\mu$-continuous vector measure $m:\mathbf {\Sigma } \to F$ of bounded variation, there exists an integrable function $f:\Omega \to F$ such that $m(S) = \int _S {f(\omega )d\mu (\omega )}$, for every $S \in \mathbf {\Sigma }$. Maynard proved that a Banach space has the Radon-Nikodym property iff it is an $s$-dentable space. It is the purpose of this paper to give the following analogous characterization: A Fréchet space $F$ has the Radon-Nikodym property iff $F$ is $s$-dentable.
References
Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 48 (1975), 371-380
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0357730-5
  • MathSciNet review: 0357730