Locally uniformly rotund renorming and fragmentability

Abstract

In this paper we characterize those Banach spaces E that admit a locally uniformly rotund renorming by means of a linear topological condition that is a particular case of the notion, introduced by J. E. Jayne, I. Namioka and C. A. Rogers, called countable cover by sets of small local diameter. This allows us to develop a Decomposition Method and a Transfer Technique to obtain locally uniformly rotund renormings. 1991 Mathematics Subject Classification: 46B20.