ISRAEL JOURNAL OF MATHEMATICS, Vol. 38, Nos. 1-2, 1981 DUAL BANACH LATTICES AND BANACH LATTICES WITH THE RADON-NIKODYM PROPERTY BY MICHEL TALAGRAND* ABSTRACT We construct a separable dual Banach lattice E such that no non-trivial order interval of its dual is weakly compact. Hence E has the Radon-Nikodym property without being in some sense a dual in a natural way. I. Introduction It has been an open problem for a long time whether every separable Banach space with the Radon-Nikodym Property (RNP -- see [2]) is a subspace of a separable dual Banach space. It is known now that the answer is negative [1], [4]. These two examples are very different, but both of them are far from being a Banach lattice. In fact, it does not seem to be even known today if a separable Banach lattice with RNP is a dual (and this question will not be answered here). An interesting idea in this direction is due to H. P. Lotz [3], along the following lines. Let E be a separable Banach lattice satisfying RNP. Let F be the set of x in the dual E* of E such that [0, Ix I] is weakly compact. Then F is a Banach lattice. Lotz shows that if F is big enough, i.e. o-(E,F) is Hausdorff, then E = F*. Hence F is a natural candidate as a predual of E. The purpose of this paper is to describe an example (whose existence is claimed in [5]) of a separable Banach lattice E (which is a dual) such that F = {0}. Hence, if E is a separable Banach lattice satisfying RNP, there does not seem to exist a natural candidate for a predual. We feel that this means both that, in general, E is not likely to be a dual, and that the problem is not likely to be easy. t The final draft of this paper was written while the author held a grant from NATO to visit the Ohio State University. Received January 30, 1980 46