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

Chi, G. Y. H.

doi:10.1090/S0002-9939-1975-0357730-5

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

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Additional Information

Journal: Proc. Amer. Math. Soc. 48 (1975), 371-380
DOI: https://doi.org/10.1090/S0002-9939-1975-0357730-5
MathSciNet review: 0357730