Strictly convex norms and topology

作者：Orihuela Jose^*; Smith Richard J; Troyanski Stanimir

来源：Proceedings of the London Mathematical Society, 2012, 104: 197-222.

DOI：10.1112/plms/pdr034

We introduce a new topological property called (*) and the corresponding class of topological spaces, which includes spaces with G(delta)-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%26apos;s compact S-space. Also, under additional axioms, we provide examples of compact scattered non-Gruenhage spaces of cardinality N-1 having (*).

• 出版日期2012-1