Hahn-Banach Separation Theorem

Theorem

Normed Vector Space

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

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:

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

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 $A$ 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:

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

and:

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

That is:

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

Hausdorff Locally Convex Space

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:

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

Source of Name

This entry was named for Hans Hahn and Stefan Banach.