On the structure of weakly compact subsets of Hilbert spaces and applications to the geometry of Banach spaces

Abstract

A characterization of weakly compact subsets of a Hilbert space, when they are considered as subsets of B B -spaces with an unconditional basis, is given. We apply this result to renorm a class of reflexive B B -spaces by defining a norm uniformly convex in every direction. We also prove certain results related to the factorization of operators. Finally, we investigate the structure of weakly compact subsets of L 1 ( μ ) {L^1}(\mu ) .