Over the separation of convex sets in a Banach space

I am working with Banach spaces and aim to prove the following property:

If $A$ and $B$ are disjoint convex sets of a Banach space $X$ with $A$ open, then $A$ and $B$ can be separated, that is, there exists a nonzero continuous real linear functional $f: X\to \Bbb R$ and a number $\alpha \in \Bbb R$ such that $$A\subset \{x | f(x)\leq \alpha\}~~\text{and}~~B\subset \{x | f(x)\geq \alpha\}.$$

I've seen this property demonstrated for locally convex topological vector spaces and I was thinking if there exist a way to prove it faster for normed spaces, but couldn't find it. What you say?

• I don't think so. It's all about convexity, having convex neighborhoods. That those neighborhoods come from a norm doesn't really help. – user357151 Jul 28 '17 at 20:41
• This is directly true for any topological vector space by the Hahn-Banach separation theorem. Are you looking for a simpler proof of Hahn-Banach on normed spaces, or is this answer sufficient? – Michael Lee Jul 29 '17 at 5:15