Generalized Metric Spaces and Locally Uniformly Rotund Renormings

Abstract

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, weak) ǁ (X,→) is A of a normedslicely continuous and Theorem 1.1 in the Introduction says that this is the case if, and only if, X has an equivalent LUR norm. After our study of this class of maps we can now formulate the following theorem and its corollaries summarizing different characterizations f LUR renormability for a Banach space.