A Nonlinear Transfer Technique for Renorming

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

“…Regarding the latter, we find an open problem [19,Problem 1.7]. Finally, assertion (vi) is related to the study of measurable selectors of upper semi-continuous multivalued functions [18].…”

“…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].…”

Are Eberlein–Grothendieck scattered spaces $$\sigma $$ σ -discrete?

“…We refer to [MOTV,Section 3.2,p.54] for information about this topic. It is easy to see that if a space with the discrete metric is SLD then it is σ-discrete and if it is σ-fragmented then it is σ-scattered.…”

Spaces of functions with countably many discontinuities

“…One of the present authors [7] recently established LUR renormability for C(K) when K is a Namioka-Phelps compact. Novel techniques developed in [11] led to a proof of LUR renormability for the space C(H), where H is the Helly space. The main result of this paper can be seen as a generalization of this last result.…”

“…The main result of this paper can be seen as a generalization of this last result. We use techniques from both [11] and [7], as well as borrowing a key idea from the recent paper [1] of Argyros et al…”