The topological complexity of a natural class of norms on Banach spaces Original Research Article

Let X be a non-reflexive Banach space such that X∗ is separable. Let View the MathML source 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 View the MathML source consisting of Fréchet-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 View the MathML source 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.