Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Complex Case/Compact Convex Set and Closed 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 compact convex set.

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

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

$\ds \sup_{x \mathop \in A} \map \Re {\map f x} < \inf_{x \mathop \in B} \map \Re {\map f x}$

Proof

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

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

$\ds \sup_{x \mathop \in A} \map g x < \inf_{x \mathop \in B} \map g x$

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:

$\ds \sup_{x \mathop \in A} \map \Re {\map f x} < \inf_{x \mathop \in B} \map \Re {\map f x}$

$\blacksquare$