Weak topologies on bounded sets of a Banach space. Associated function spaces.

Abstract

This is a survey paper concerning:\par 1) relations between spaces of weakly continuous functions on Banach spaces and their weak topologies;\par 2) weakly continuous and weakly differentiable function spaces in relation with the extension of Weierstrass' theorem for infinite dimensional Banach spaces;\par 3) interpolation of bounded sequences by weakly continuous and weakly differentiable functions.