Let $(E,\tau)$ be a locally convex $\mathbb{K}$-vector-space and $A\subseteq E$ with $0\in A$. The following statements are equivalent:
(1) $A$ is closed and convex
(2) It exists a subset $B\subseteq E'$ ($E'$ notes the dual space of $E$) with $A=\bigcap_{f\in B}\{\Re(f(x))\leq 1\}$
As a hint it is given, that one should first show, that for every $x\in E\setminus A$ there is an open convex neighborhood $U$ of $x$ such that $U\cap A=\emptyset$
I think this is related to the seperation theorem of Hahn-Banach.
I have no idea on how to proof the hint. :( Can you get me started?
Thanks in advance.