Math400 - Functional Analysis - Section 4.2 - The weak topology of a normed space - Part 3

Convex sets and linear operators in the weak topology. The weak topology as a vector space topology

[cristianjulianduarterangel]

Do you know any application of Mazur's Corollary? When I saw it the first time, I felt it was just a convergence transport from the weak topology to the strong one, with the convex hull "in the middle". But I would like to understand it beyond that.