On Weakly Locally Uniformly Rotund Banach Spaces

We show that every normed spaceEwith a weakly locally uniformly rotund norm has an equivalent locally uniformly rotund norm. After obtaining aσ-discrete network of the unit sphereSEfor the weak topology we deduce that the spaceEmust 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 descriptive Banach spaces studied by Hansell, so we present here some new characterizations of them.