TITLE: Convex set and weak topology
QUESTION [7 upvotes]: i have this question and i don't know how to answer it 
"Let $E$ be a Banach space. Let $A \subset E$ be a convex subset. Prove that the closure
of $A$ in the strong topology and that in the weak topology $\sigma (E,E^*)$ are the same"
Help me please ,thank you.

REPLY [6 votes]: We can prove it by double inclusion.
One direction follows from the fact that the weak topology has less open sets than the topology induced by the norm. 
For the other direction, we can use the geometric form of Hahn-Banach theorem.