On Weakly Locally Uniformly Rotund Banach Spaces

Abstract

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

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

“…It is worth noting that the notions which appear in the former result are closely related to LUR renorming [16,17,18]. Property SLD was introduced and studied by Jayne, Namioka, and Rogers [10], the notion of σ-isolated family was introduced by Hansell [6], and the class of maps with a σ-isolated function base was studied by Hansell [8].…”

“…Let us begin this section by showing some useful results. The first one is an equivalence in [17,Proposition 2]. Here, we will provide a direct proof in order to have a self-contained work.…”

“…Characterizations for spaces having property SLD can be found in [17]. We begin this section by providing two results in this aim.…”

Descriptive compact spaces and renorming

“…Descriptive Banach spaces have been studied in [12,21] and also in the context of renorming theory in [18,19,[23][24][25].…”

“…Some results have recently been obtained showing that geometrical properties such as the existence of equivalent Kadec or locally uniformly rotund (LUR) norms in a Banach space X can be characterized by the existence of certain types of networks of the norm topology which are σ-isolated for the weak topology of X (LUR norms [18,19,24], dual LUR norms [24,25] and Kadec norms [23]). Recently, in [8], it has been proved that the dual unit ball (with its weak * topology) is uniformly Eberlein if, and only if, the dual space has a w * -UR equivalent norm, which is equivalent to X having a uniformly Gateaux smooth equivalent norm.…”