A geometrical characterization of Banach spaces with the Radon-Nikodym property

Abstract

A characterization of Banach spaces having the Radon-Nikodym proper- ty is obtained in terms of a convexity requirement on all bounded subsets. In addition a Radon-Nikodym theorem, utilizing this convexity property, is given for the Bochner integral and it is easily shown that this theorem is equivalent to the Phillips-Metivier Radon-Nikodym theorem as well as all the standard Radon-Nikodym theorems for the Bochner integral. l. Introduction. Rieffel (9) proved a Radon-Nikodym theorem for the Bochner integral, using techniques established in (8), in an attempt to establish the Radon- Nikodym theorem of Phillips (7) and Metivier (5). He was unable to establish it in the nonseparable case, the result depending upon a proof that every convex weakly compact set in a B-space is dentable. This circle of ideas was not closed until Troyanski (10) proved that a Banach space with a weakly compact fundamental subset is isomorphic to a locally uniformly convex Banach space. This is, as would be expected, much deeper than necessary and a simpler proof will be indicated in ?2. The obvious characterization of Banach spaces with the Radon-Nikodym property would seem to be that every bounded subset must be dentable. In ?3 it is demonstrated that a characterization is that every bounded subset must be o- dentable, where a-dentability is a dentable type condition which is strictly weaker than dentability. It is however an open question if dentable and a-dentable coincide in Banach spaces having the Radon-Nikodym property.