Let X be a non-reflexive Banach space such that X ∗ is separable. Let N(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 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 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.