Topological Modules/Hahn–Banach theorems

Theorem (geometrical Hahn–Banach theorem):

Let ${\displaystyle V}$ be a real topological vector space, and let ${\displaystyle U\subseteq V}$ be open and convex so that ${\displaystyle 0\notin V}$. Then there exists a hyperplane ${\displaystyle W\leq V}$ not intersecting ${\displaystyle U}$.

(On the condition of the axiom of choice.)

Proof: The set of all vector subspaces of ${\displaystyle V}$ that do not intersect is inductive and also nonempty (because of the zero subspace). Hence, by Zorn's lemma, pick a maximal vector subspace ${\displaystyle W\leq V}$ that does not intersect ${\displaystyle U}$. Claim that ${\displaystyle W}$ is a hyperplane. If not, ${\displaystyle V/W}$ has dimension ${\displaystyle \geq 2}$. Now the canonical map ${\displaystyle p:V\to V/W}$ is open, so that ${\displaystyle U':=p(U)}$ is an open, convex subset of ${\displaystyle V/W}$. We consider the cone

${\displaystyle C:=\bigcup _{\lambda >0}\lambda U'}$

and note that it has a nonzero boundary point; for otherwise ${\displaystyle C}$ would be clopen in ${\displaystyle V/W\setminus \{0\}}$ which is path-connected (indeed by assumption ${\displaystyle \operatorname {dim} V/W\geq 2}$, so that for any two points ${\displaystyle x,y\in V/W}$ we find a 2-dimensional plane containing both, and by using a "corner point" ${\displaystyle z}$ when ${\displaystyle x,y}$ do lie on a line through the origin, we may connect them in ${\displaystyle V/W\setminus \{0\}}$, because a segment in a TVS yields a continuous path by continuity of addition and scalar multiplication), so that ${\displaystyle C=V/W\setminus \{0\}}$, which is impossible because for any ${\displaystyle x\neq 0}$ in ${\displaystyle V/W}$, we then have ${\displaystyle x\in \mu U'}$, ${\displaystyle -x\in \lambda U'}$ for ${\displaystyle \mu ,\lambda >0}$, so that ${\displaystyle 0\in U'}$ by convexity, a contradiction. Hence, let ${\displaystyle w\in \partial C}$. Then the line ${\displaystyle L}$ generated by ${\displaystyle w}$ does not intersect ${\displaystyle C}$ and hence not ${\displaystyle U'}$, and ${\displaystyle p^{-1}(L)}$ is a larger subspace of ${\displaystyle V}$ that does not intersect ${\displaystyle U}$ than ${\displaystyle W}$ in contradiction to the maximality of the latter. ${\displaystyle \Box }$