The topological complexity of a natural class of norms on Banach spaces

Let X be a non-reflexive Banach space such that X* is separable. Let X(X) be the set of all equivalent norms on X, equipped with the topology of uniform convergence on bounded subsets of X. We show that the subset Z of N(X) consisting of Frechet-differentiable norms whose dual norm is not strictly convex reduces any difference of analytic sets. It follows that Z is exactly a difference of analytic sets when. N(X) is equipped with the standard Effros-Borel structure. Our main lemma elucidates the topological structure of the norm-attaining linear forms when the norm of X is locally uniformly rotund. (C) 2001 Elsevier Science B.V. All rights reserved.