The Radon-Nikodym property and dentable sets in Banach spaces

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 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 E is s 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.