Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Complex Case/Open Convex Set and Convex Set

Theorem
Let $\struct {X, \PP}$ be a Hausdorff locally convex space over $\C$ equipped with its standard topology.

Let $X^\ast$ be the topological dual space of $\struct {X, \PP}$. Let $A \subseteq X$ be an open convex set.

Let $B \subseteq X$ be a convex set disjoint from $A$.

Then there exists $f \in X^\ast$ and $c \in \R$ such that:


 * $A \subseteq \set {x \in X : \map \Re {\map f x} < c}$

and:


 * $B \subseteq \set {x \in X : \map \Re {\map f x} \ge c}$

That is:


 * there exists $f \in X^\ast$ and $c \in \R$ such that $\map \Re {\map f a} < c \le \map \Re {\map f b}$ for each $a \in A$ and $b \in B$.

Proof
Let $X_\R$ be the realification of $X$.

Applying Hahn-Banach Separation Theorem: Hausdorff Locally Convex Space: Real Case: Open Convex Set and Convex Set, there exists a continuous $\R$-linear functional $g : X \to \R$ and $c \in \R$ such that:


 * $\map g a < c \le \map g b$ for each $a \in A$ and $b \in B$.

From Continuous Real Linear Functional on Complex Topological Vector Space is Real Part of Continuous Complex Linear Functional, there exists $f \in X^\ast$ such that:


 * $\map g x = \map \Re {\map f x}$ for each $x \in X$.

Then, we have:


 * $\map \Re {\map f a} < c \le \map \Re {\map f b}$ for each $a \in A$ and $b \in B$.