The Radon-Nikodym property and dentable sets in Banach spaces

Authors:
W. J. Davis and R. R. Phelps

Journal:
Proc. Amer. Math. Soc. **45** (1974), 119-122

MSC:
Primary 46B05

DOI:
https://doi.org/10.1090/S0002-9939-1974-0344852-7

MathSciNet review:
0344852

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Abstract | References | Similar Articles | Additional Information

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 ``-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 is -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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0344852-7

Keywords:
Banach space,
dentable sets,
vector valued measures,
Radon-Nikodym theorem

Article copyright:
© Copyright 1974
American Mathematical Society