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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.

To motivate our results we recall the following theorem of Bishop and Phelps [_3] : Theorem. Let D be a closed, bounded and convex subset of a Banach space X. Then the set of support functionals of D (that is, the set of elements of the dual X* of X that attain their suprema on D) is a norm dense subset of X*.Phrasing this another way, if x* is any element of X* and 6 is any positive number then there exists a y* in X* with ily*H 0, there exists a continuous, linear function T:X~ Y of rank one with [I TI[ <6 and such that f+ Tattains its supremum on D. We give a complete answer to this problem. Basically, the most important thing is the set D. If it satisfies a certain property (what we call an RNP set) thenfmay be any reasonable mapping and Yarbitrary. Conversely, if D is not well-behaved (in our sense) thenfmay be very nice (for example, continuous and convex), Y the real numbers, and f will not have close, norm attaining approximations even in a sense more general than that of (Q).As examples of the sort of results we obtain let us recall the following theorem of Lindenstrauss [8] : Theorem. Let X and Y be Banach spaces, W a weakly compact subset of X, and let 5~ (X, I1) denote the space of continuous, .linear functions from X to Y with the usual operator norm. Then the set of elements of ,~(X, Y) that attain their suprema in norm on W is a norm dense subset of f(X, Y).Recently, in a remarkable result Bourgain [5] has shown that for RNP sets (definition below) the characterization is exact: Theorem 1. Let D be a closed, bounded, convex and symmetric subset of the Banach space X. Then D is an RNP set if and only iffor every Banach space ~ the subset of

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, 2, ju) and any X valued measure m on E, with m having finite total variation and being absolutely continuous with respect to fi, then m is the indefinite integral with respect to ß of an X valued Bochner integrable function on S. The first study of this property

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