The topological complexity of a natural class of norms on Banach spaces (Q5942880)

Let \(X\) be a non-reflexive Banach space such that \(X^*\) is separable. Denote by \({\mathcal N}(X)\) the set of equivalent norms on \(X\), equipped with the topology of uniform convergence on bounded subsets of \(X\). Let \(Z\) be the subset of \({\mathcal N}(X)\) consisting the Fréchet-differentiable norms whose dual norm is not strictly convex. It is shown that \(Z\) reduces any difference of analytic sets. \(Z\) is exactly a difference of analytic sets when \({\mathcal N}(X)\) is equipped with the standard Effros-Borel structure.