Generalized Metric Spaces and Locally Uniformly Rotund Renormings

A class of generalized metric spaces is a class of spaces defined by a property shared by all metric αspaces which is close to metrizability in some sense [Gru84]. The s-spaces are defined by replacing the base by network in the Bing-Nagata-Smirnov metrization theorem; i.e. a topological space is a αspace if it has a αdiscrete network. Here we shall deal with a further re- finement replacing discrete by isolated or slicely isolated. Indeed we will see that the identity map from a subset A of a normed space is A of a normedslicely continuous if, and only if, the weak topology relative to A has a s-slicely isolated network. If A is also a radial set then we have that the identity map Id : (X,…

Some Open Problems

We have extensively considered here the use of Stone's theorem on the paracompactness of metric spaces in order to build up new techniques to construct an equivalent locally uniformly rotund norm on a given normed space X. The discreetness of the basis for the metric topologies gives us the necessary rigidity condition that appears in all the known cases of existence of such a renorming property [Hay99, MOTV06]. Our approximation process is based on co-σ-continuous maps using that they have separable fibers, see Sect. 2.2. We present now some problems that remain open in this area. Some of them are classical and have been asked by different authors in conferences, papers and books. Others h…

Locally uniformly rotund renorming and fragmentability

σ-Slicely Continuous Maps

All examples of σ-slicely continuous maps are connected somehow with LUR Banach spaces. It is clear that if x is a denting point of a set D and Φ is a norm continuous map at x then Φ is slicely continuous at x. Hence if X is a LUR normed space then every norm continuous map Φ on B X is slicely continuous on S X .

Kadec and Krein–Milman properties

Abstract The main goal of this paper is to prove that any Banach space X with the Krein–Milman property such that the weak and the norm topology coincide on its unit sphere admits an equivalent norm that is locally uniformly rotund.

σ-Continuous and Co-σ-continuous Maps

In this chapter we isolate the topological setting that is suitable for our study. We first present 2.1–2.3 to follow an understandable logical scheme nevertheless the main contribution are presented in 2.4–2.7 and our main tool will be Theorem 2.32. An important concept will be the σ-continuity of a map Φ from a topological space (X, T) into a metric space (Y, g). The σ-continuity property is an extension of continuity suitable to deal with countable decompositions of the domain space X as well as with pointwise cluster points of sequences of functions Φn : X → Y, n = 1,2,… When (X,T) is a subset of a locally convex linear topological space we shall refine our study to deal with σ-slicely …

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 σ -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 descript…