Topological Modules/Hahn–Banach theorems

Theorem (geometrical Hahn–Banach theorem):

Let be a real topological vector space, and let be open and convex so that . Then there exists a hyperplane not intersecting .

(On the condition of the axiom of choice.)

Proof: The set of all vector subspaces of that do not intersect is inductive and also nonempty (because of the zero subspace). Hence, by Zorn's lemma, pick a maximal vector subspace that does not intersect . Claim that is a hyperplane. If not, has dimension . Now the canonical map is open, so that is an open, convex subset of . We consider the cone

and note that it has a nonzero boundary point; for otherwise would be clopen in which is path-connected (indeed by assumption , so that for any two points we find a 2-dimensional plane containing both, and by using a "corner point" when do lie on a line through the origin, we may connect them in , because a segment in a TVS yields a continuous path by continuity of addition and scalar multiplication), so that , which is impossible because for any in , we then have , for , so that by convexity, a contradiction. Hence, let . Then the line generated by does not intersect and hence not , and is a larger subspace of that does not intersect than in contradiction to the maximality of the latter.