# Geometric version of Hahn-Banach theorem

How to prove the following geometric version of Hahn-Banach theorem:

Let $A$ and $B$ be nonempty open disjoint convex subsets of a normed linear space $E$. Then there exist a nonzero $f \in E^*$ and $\alpha \in \mathbb{R}$ such that $A \subset \{x \in E : f(x) > \alpha\}$ and $B \subset \{x \in E : f(x) < \alpha\}$.

• A general hint. To separate $A$ from $B$, we can separate $A - B$ from $\{0\}$. – GEdgar Mar 16 '16 at 11:42