The Radon-Nikodym property in conjugate Banach spaces

Abstract

We characterize conjugate Banach spaces X* having the Radon-Nikodym Property as those spaces such that any separable subspace of X has a separable conjugate. Several applications are given. Introduction. There are several equivalent formulations of the Radon-Nikodym Property (RNP) in Banach spaces; we give perhaps the earliest definition: a Banach space X has RNP if given any finite measure space (S, E, p) and any X valued measure m on E, with m having finite total variation and being absolutely continuous with respect to ,, then m is the indefinite integral with respect to ,u of an X valued Bochner integrable function on S. The first study of this property was by Dunford and Pettis [4] and Phillips [11] (see also [5]). It follows from the work of Dunford and Pettis and Phillips that reflexive Banach spaces and separable conjugate spaces have RNP. More generally, the following is true: THEOREM A. If X is a Banach space such that for any separable subspace Y of X, Y* is separable, then X* has RNP. The above result was observed by Ubl [15] and also can be obtained from a result of Grothendieck (Theorem B below). The first characterizations of RNP were given by Grothendieck in [6]. Grothendieck's approach, the one we shall use, is that of studying certain classes of operators. An operator T: X -+ Y is a continuous linear function T from the Banach space X to the Banach space Y. An operator T: X -> Y is said to be an integral operator if there exist a compact Hausdorff space K, a Radon measure ,u on K, and operators R, and S, such that Received by the editors July 24, 1973 and, in revised form, February 2, 1974. AMS (MOS) subject classifications (1970). Primary 28A45, 46G10, 46B99.