Strictly convex norms and topology

Abstract

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 characterise those Banach spaces which admit equivalent strictly convex norms, and give an internal topological characterisation of those scattered compact spaces $K$, for which the dual Banach space $C(K{)}^{\ast }$ 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 $C(\mathcal{K}{)}^{\ast }$ admits an equivalent strictly convex dual norm, where $\mathcal{K}$ is Kunen's compact space. Also, under the continuum hypothesis CH, we give an example of a compact scattered non-Gruenhage space having (*).

References in corpus (2)

• Gruenhage compacta and strictly convex dual norms
• Unconditional bases and strictly convex dual renormings

Cited by in corpus (4)

• Generalized metric properties of spheres and renorming of normed spaces
• Chains of functions in $C(K)$-spaces
• A topological characterization of dual strict convexity in Asplund spaces
• Tree duplicates, $G_{\delta }$-diagonals and Gruenhage spaces