Separation sentence

The separation theorem (also theorem of Eidelheit , named after Meier Eidelheit ) is a mathematical theorem about the possibilities of separating convex sets in normalized vector spaces (or more generally locally convex spaces ) by linear functionals . These are geometrical inferences from Hahn-Banach's theorem . Like this, the separation theorem is based on a non- constructive argument, such as the lemma of Zorn or the axiom of choice .

First formulation

The simplest version of the separation theorem is as follows:

Let be a normalized vector space (or locally convex space ) over or . Let further be a closed convex set and . Then there exists a linear continuous functional with ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle M\subset X}$ ${\displaystyle x\in X\setminus M}$ ${\displaystyle \varphi \in X'}$

${\displaystyle {\rm {Re}}(\varphi (x))<\inf\{{\rm {Re}}(\varphi (y))\ |\ y\in M\}}$.

Here denotes the real part and the topological dual space of . One then says: The functional separates the point from the set . ${\displaystyle {\rm {Re}}}$ ${\displaystyle X'}$ ${\displaystyle X}$ ${\displaystyle \varphi }$ ${\displaystyle x}$ ${\displaystyle M}$

Further formulations

In the above formulation, the point can be replaced by a compact convex set. We then get the following theorem: ${\displaystyle x}$

Let be a normalized vector space (or locally convex space) over or . Let further be a non-empty, closed, convex set and a non-empty, compact, convex set. Then there exists a linear continuous functional with ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle M\subset X}$ ${\displaystyle K\subset X\setminus M}$ ${\displaystyle \varphi \in X'}$

${\displaystyle \sup\{{\rm {Re}}(\varphi (x))\ |\ x\in K\}<\inf\{{\rm {Re}}(\varphi (y))\ |\ y\in M\}}$.

Ultimately, one comes to a weaker separation property if one renounces the closure and compactness in the above version:

Let be a normalized vector space (or locally convex space) over or . Further be non-empty, disjoint, convex sets, be open . Then there exists a linear continuous functional with ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle M_{1},M_{2}\subset X}$ ${\displaystyle M_{2}}$ ${\displaystyle \varphi \in X'}$

${\displaystyle \forall y\in M_{2}:\sup\{{\rm {Re}}(\varphi (x))\ |\ x\in M_{1}\}<{\rm {Re}}(\varphi (y))}$.

Hyperplanes

In the visual space, disjoint convex sets are separated by planes. ${\displaystyle {\mathbb {R} }^{3}}$

Sets of the form , where and , are closed hyperplanes . They divide the room into an upper half-space and a lower half-space . For a compact convex set and a disjoint, closed convex set, one can find a hyperplane according to the above separation theorem, so that the two sets lie in different half-spaces, namely in the interior of these half-spaces. It is said that the hyperplane separates the two convex sets. This is particularly clear in the 2-dimensional and 3-dimensional case, since the hyperplanes in these cases are straight lines or planes . ${\displaystyle \{x\in X;{\rm {Re}}(\varphi (x))=r\}}$ ${\displaystyle \varphi \in X'}$ ${\displaystyle r\in {\mathbb {R} }}$ ${\displaystyle X}$ ${\displaystyle \{x\in X;{\rm {Re}}(\varphi (x))\geq r\}}$ ${\displaystyle \{x\in X;{\rm {Re}}(\varphi (x))\leq r\}}$

The disjoint, convex sets and cannot be separated by open half-spaces. ${\displaystyle M_{1}}$ ${\displaystyle M_{2}}$

If one has two disjoint convex sets in , one of which is open, according to the last-mentioned version of the separation theorem, there is also a hyperplane for them, so that the two sets lie in different half-spaces. In general, however, it can no longer be achieved that both are inside the half-spaces. To do this, consider the lower half-plane and the open set above the graph of the exponential function . As illustrated by adjacent drawing, with the only separating hyperplane, and not within the interior of the associated half-space. ${\displaystyle X}$ ${\displaystyle X={\mathbb {R} }^{2}}$ ${\displaystyle M_{1}:=\{(x,y)\ |\ y\leq 0\}}$ ${\displaystyle M_{2}:=\{(x,y)\ |\ y>e^{x}\}}$ ${\displaystyle \{(x,y)\ |\ \varphi (x,y)=0\}}$ ${\displaystyle \varphi (x,y):=y}$ ${\displaystyle M_{1}}$

Applications

This theorem has many important applications outside of functional analysis and represents a non- constructive existential argument for many proofs , including:

• Existence of subdifferentials for suitably formulated generalized directional derivatives .
• Proof of Farkas' lemma , that is, application in convex optimization .
• Proof of the fundamental theorem of arbitrage price theory for fair prices of derivatives in the multi-period model.

literature

• Richard Kadison , John Ringrose : Fundamentals of the Theory of Operator Algebras (Graduate studies in mathematics; 15/16). American Mathematical Society, Providence, RI 1997 (EA 1983)

1. 1997, ISBN 0-8218-0819-2 .
2. 1997, ISBN 0-8218-0820-6 .

• Dirk Werner : Functional Analysis . 7th edition. Springer, Berlin 2011, ISBN 978-3-642-21016-7 (EA Berlin 1995)