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?