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 $B \subset \{x \in E : f(x) < \alpha\}$.

Thank you for your help.

  • 1
    $\begingroup$ A general hint. To separate $A$ from $B$, we can separate $A - B$ from $\{0\}$. $\endgroup$ – GEdgar Mar 16 '16 at 11:42

A complete proof (with multiple lemmas) is given in

Haïm Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, p. 4-8.

  • $\begingroup$ What happens when the sets are closed, or empty, or non-convex? $\endgroup$ – Octagonal Monk Jun 23 at 12:48

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