Kerin's continuation

The continuation of Kerin ( English Krein's extension theorem ) is a doctrine of the mathematical branch of analysis , which goes back to a work presented in 1937 by the Soviet mathematician Mark Grigorjewitsch Kerin (1907-1989) . The Kerin continuation theorem gives an answer to the question under which conditions continuations of positive linear functionals on real vector spaces are possible, and in this respect is related to (and can even be derived from) Hahn-Banach's theorem . Like this and other continuation theorems of mathematics, his proof is based on the lemma of Zorn and thus requires the acceptance of the validity of the axiom of choice .

Formulation of the sentence

The continuation of Kerin occurs in two - but closely related - formulations.

Formulation according to Neumark

The Soviet mathematician Mark Neumark presented one formulation of the continuation sentence in his monograph Normated Algebras :

Given a locally convex topological - vector space and is a non-empty convex cone and a linear subspace . ${\displaystyle \mathbb {R} }$ ${\displaystyle E}$ ${\displaystyle P\subseteq E}$ ${\displaystyle S\subseteq E}$

The cone should contain inner points and should be valid, i.e. at least one point should also be the point of the subspace . ${\displaystyle P}$ ${\displaystyle \left(S\cap P^{\circ }\right)\neq \emptyset }$ ${\displaystyle p\in P^{\circ }}$ ${\displaystyle S}$

Then:

Every positive linear functional defined on the subspace can be continued into a positive linear functional defined over the entire space . ${\displaystyle S}$ ${\displaystyle E}$

That means: If there is a linear functional that satisfies the condition for all , then there is always a functional with for all and for all . ${\displaystyle f\colon S\rightarrow \mathbb {R} }$ ${\displaystyle f(p)\geq 0}$ ${\displaystyle p\in \left(S\cap P\right)}$ ${\displaystyle F\colon E\rightarrow \mathbb {R} }$ ${\displaystyle F(p)\geq 0}$ ${\displaystyle p\in P}$ ${\displaystyle F(x)=f(x)}$ ${\displaystyle x\in S}$

Formulation according to Hewitt / Stromberg

A somewhat different formulation of the continuation of Kerin can be found in the monograph Real and Abstract Analysis by the two American mathematicians Edwin Hewitt and Karl Robert Stromberg :

Given a -vector space and in it a non-empty convex cone and a linear subspace . ${\displaystyle \mathbb {R} }$ ${\displaystyle E}$ ${\displaystyle P\subseteq E}$ ${\displaystyle S\subseteq E}$

With regard to the relationships between the cone and the secondary classes of the subspace, it should apply that a point satisfies the condition if and only if the corresponding condition is given for the mirror point . ${\displaystyle P}$ ${\displaystyle S}$ ${\displaystyle x\in E}$ ${\displaystyle \left(\left[x+S\right]\cap P\right)\neq \emptyset }$ ${\displaystyle -x}$ ${\displaystyle \left(\left[-x+S\right]\cap P\right)\neq \emptyset }$

Then:

A positive linear functional defined on the subspace can always be continued to a positive linear functional defined on the entire space . ${\displaystyle S}$ ${\displaystyle E}$

Immediate inference

From Kerin's continuation the following sentence is drawn as a direct consequence:

If a convex cone is in a locally convex topological vector space and if there is an inner point located in it, there is always a positive linear functional with . ${\displaystyle P}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle E}$ ${\displaystyle p_{0}\in P^{\circ }}$ ${\displaystyle \phi \colon E\rightarrow \mathbb {R} }$ ${\displaystyle {\phi }(p_{0})=1}$

annotation

Hewitt and Stromberg explicitly refer to the Kerin continuation clause as Krein's extension theorem for non-negative linear functionals . In this context, it should be noted that in the analytical specialist literature, instead of non-negative linear functionals (or similar), it is not uncommon to also speak of positive linear functionals (or similar). Are meant in each case real valued linear functionals on the given topological vector space that induced by the convex cone order structure monotonously in the order of structure transmitted. ${\displaystyle \mathbb {R} }$

literature

• Edwin Hewitt, Karl R. Stromberg: Real and Abstract Analysis: A Modern Treatment of the Theory of Functions of a Real Variable (=  Graduate Texts in Mathematics . Volume 25 ). 3. Edition. Springer-Verlag , New York, Heidelberg, Berlin 1975, ISBN 0-387-90138-8 ( MR0367121 ).
• MG Kerin: About positive additive functionals in linear normalized spaces ( Ukrainian ) . In: Comm. Soc. Math. Kharkov . tape 14 , 1937, pp. 227-237 .
• MG Kerin, MA Rutman : Linear operators leaving invariant a cone in a Banach space . In: Amer. Math. Soc. Translation . tape 1950 , 1950 ( MR0038008 ).
• Mark Neumark: Normalized Algebras . Verlag Harri Deutsch , Thun and Frankfurt / Main 1990, ISBN 3-8171-1001-4 .

Individual evidence

1. ^ ^{A b} Edwin Hewitt, Karl R. Stromberg: Real and Abstract Analysis. 1975, pp. 219-220
2. ^ Mark Neumark: Normalized Algebras. 1990, pp. 84-85
3. ^ Neumark, op.cit., P. 84
4. ^ Neumark, op.cit., P. 85
5. ↑ Hewitt / Stromberg, op.cit., P. 220