On weakly locally uniformly rotund Banach spaces

We show that every normed space E with a weakly locally uniformly rotund no rm has an equivalent locally uniformly rotund norm. After obtaining a sigma -discrete network of the unit sphere S-E for the weak topology we deduce th at 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 pos ed by Deville, Godefroy, Haydon, and Zizler. For a weakly uniformly rotund norm we prove that the unit sphere is always metrizable for the weak topolo gy despite the fact that it may not have the Kadec property. Moreover, Bana ch 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. (C) 1999 Academic Press.