Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space
Theorem
Real Case
Let $\struct {X, \PP}$ be a Hausdorff locally convex space over $\R$ equipped with its standard topology.
Let $X^\ast$ be the topological dual space of $\struct {X, \PP}$.
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:
- $A \subseteq \set {x \in X : \map f x < c}$
and:
- $B \subseteq \set {x \in X : \map f x \ge c}$
That is:
- there exists $f \in X^\ast$ and $c \in \R$ such that $\map f a < c \le \map f b$ for each $a \in A$ and $b \in B$.
Compact Convex Set and Closed Convex Set
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:
- $\ds \sup_{x \mathop \in A} \map f x < \inf_{x \mathop \in B} \map f x$
Complex Case
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 $\struct {X, \PP}$.
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 $A \subseteq X$ 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:
- $\ds \sup_{x \mathop \in A} \map \Re {\map f x} < \inf_{x \mathop \in B} \map \Re {\map f x}$
Source of Name
This entry was named for Hans Hahn and Stefan Banach.
Also known as
These theorems are sometimes known as the geometric Hahn-Banach theorems.