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We introduce a new topological property called (*) and the corresponding class of topological spaces, which includes spaces with Gδ‐diagonals and Gruenhage spaces. Using (*), we characterize those Banach spaces which admit equivalent strictly convex norms, and give an internal topological characterization of those scattered compact spaces K, for which the dual Banach space C(K)* admits an equivalent strictly convex dual norm. We establish some relationships between (*) and other topological concepts, and the position of several well‐known examples in this context. For instance, we show that C(𝒦)* admits an equivalent strictly convex dual norm, where 𝒦 is Kunen's compact S‐space. Also, under additional axioms, we provide examples of compact scattered non‐Gruenhage spaces of cardinality ℵ1 having (*).

We show that every normed space E with a weakly locally uniformly rotund norm has an equivalent locally uniformly rotund norm. After obtaining a _-discrete network of the unit sphere S E for the weak topology we deduce that the space E must have a countable cover by sets of small local diameter, which in turn implies the renorming conclusion. This solves a question posed by Deville, Godefroy, Haydon, and Zizler. For a weakly uniformly rotund norm we prove that the unit sphere is always metrizable for the weak topology despite the fact that it may not have the Kadec property. Moreover, Banach spaces having a countable cover by sets of small local diameter coincide with the descriptive Banach spaces studied by Hansell, so we present here some new characterizations of them. Academic Press