Hahn-Banach Separation Theorem/Normed Vector Space/Complex Case

Theorem

Let $\struct {X, \norm \cdot}$ be a normed vector space over $\C$.

Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual space of $\struct {X, \norm \cdot}$.

Open Convex Set and Convex Set

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 $b \in B$.

Compact Convex Set and Closed Convex Set

Let $A$ be a compact convex set.

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

Then there exists $f \in X^\ast$, $c \in \R$ and $\epsilon > 0$ such that:

$A \subseteq \set {x \in X : \map \Re {\map f x} \le c - \epsilon}$

and:

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

That is:

there exists $f \in X^\ast$, $c \in \R$ and $\epsilon > 0$ such that $\map \Re {\map f a} \le c - \epsilon < c + \epsilon \le \map \Re {\map f b}$ for $a \in A$ and $b \in B$.