The Radon-Nikodym property and dentable sets in Banach spaces

Davis, W. J.; Phelps, R. R.

doi:10.1090/S0002-9939-1974-0344852-7

The Radon-Nikodym property and dentable sets in Banach spaces
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by W. J. Davis and R. R. Phelps PDF

Proc. Amer. Math. Soc. 45 (1974), 119-122 Request permission

Abstract:

In order to prove a Radon-Nikodym theorem for the Bochner integral, Rieffel [5] introduced the class of “dentable” subsets of Banach spaces. Maynard [3] later introduced the strictly larger class of “$s$-dentable” sets, and extended Rieffel’s result to show that a Banach space has the Radon-Nikodym property if and only if every bounded nonempty subset of $E$ is $s$-dentable. He left open, however, the question as to whether, in a space with the Radon-Nikodym property, every bounded nonempty set is dentable. In the present note we give an elementary construction which shows this question has an affirmative answer.

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