WEAKLY METRIZABILITY OF SPHERES AND RENORMINGS OF BANACH SPACES.

We study the metrizability of weak topologies when restricted to the unit sphere of some equivalent norm on a Banach space, and its relationships with other geometrical properties of norms. In case of dual Banach space X*, we prove that there exists a dual norm such that its unit sphere is weak* metrizable if and only if (B_X*, w*) is a descriptive compact, which provides a complete characterization. [ABSTRACT FROM AUTHOR]