Strictly convex norms and topology

Abstract

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 (*).

A topological characterization of dual strict convexity in Asplund spaces

“…We take a brief detour to explore the relationship between the functions ϕ and so-called F -distance. We feel that this detour is justified, given the clear connections between this paper and [6,7], wherein F -distance is introduced and put to use. We define a seminorm on X * * by…”

“…Then C(K) * admits an equivalent strictly convex dual norm e.g. by Theorem 1.3 (though we stress that this fact was known long before the appearance of [6]). However, since K is not metrizable, and embeds homeomorphically inside (S C(K) * , w * ), taken with respect to the canonical norm, it follows that the latter space does not have a G δ -diagonal.…”

“…There has been a large collective effort on the part of a number of mathematicians to find such characterizations. The reader is referred to the Introduction of [6] for an account of this journey. In more recent years, concerning a number of questions in renorming theory in general, it has been recognised that effective characterizations can be given, at least in part, using the language of topology.…”

“…The rest of the paper is organised as follows. The next section introduces a class of convex functions that is needed in the proof of Theorem 1.4 (4) ⇒ (1), and relates these functions to the notion of F -distance, which was used in [6,7]. The proof of Theorem 1.4 (4) ⇒ (1) is given in Section 3.…”

On Gruenhage spaces, separating $$\sigma $$ σ -isolated families, and their relatives

“…The latter was introduced by Gruenhage [4] in order to solve a problem stated in [32]. Gruenhage spaces have already been studied in [3,21,27,28,29,30,31].…”