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A geometric characterization of Fréchet spaces with the Radon-Nikodým property


Author: G. Y. H. Chi
Journal: Proc. Amer. Math. Soc. 48 (1975), 371-380
DOI: https://doi.org/10.1090/S0002-9939-1975-0357730-5
MathSciNet review: 0357730
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Abstract | References | Additional Information

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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0357730-5
Keywords: Fréchet spaces, Radon-Nikodym property, $ s$-dentability, dentability
Article copyright: © Copyright 1975 American Mathematical Society

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