M. Riesz extension theorem

The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz[1] during his study of the problem of moments.[2]

Formulation

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Let   be a real vector space,   be a vector subspace, and   be a convex cone.

A linear functional   is called  -positive, if it takes only non-negative values on the cone  :

 

A linear functional   is called a  -positive extension of  , if it is identical to   in the domain of  , and also returns a value of at least 0 for all points in the cone  :

 

In general, a  -positive linear functional on   cannot be extended to a  -positive linear functional on  . Already in two dimensions one obtains a counterexample. Let   and   be the  -axis. The positive functional   can not be extended to a positive functional on  .

However, the extension exists under the additional assumption that   namely for every   there exists an   such that  

Proof

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The proof is similar to the proof of the Hahn–Banach theorem (see also below).

By transfinite induction or Zorn's lemma it is sufficient to consider the case dim  .

Choose any  . Set

 

We will prove below that  . For now, choose any   satisfying  , and set  ,  , and then extend   to all of   by linearity. We need to show that   is  -positive. Suppose  . Then either  , or   or   for some   and  . If  , then  . In the first remaining case  , and so

 

by definition. Thus

 

In the second case,  , and so similarly

 

by definition and so

 

In all cases,  , and so   is  -positive.

We now prove that  . Notice by assumption there exists at least one   for which  , and so  . However, it may be the case that there are no   for which  , in which case   and the inequality is trivial (in this case notice that the third case above cannot happen). Therefore, we may assume that   and there is at least one   for which  . To prove the inequality, it suffices to show that whenever   and  , and   and  , then  . Indeed,

 

since   is a convex cone, and so

 

since   is  -positive.

Corollary: Krein's extension theorem

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Let E be a real linear space, and let K ⊂ E be a convex cone. Let x ∈ E/(−K) be such that R x + K = E. Then there exists a K-positive linear functional φE → R such that φ(x) > 0.

Connection to the Hahn–Banach theorem

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The Hahn–Banach theorem can be deduced from the M. Riesz extension theorem.

Let V be a linear space, and let N be a sublinear function on V. Let φ be a functional on a subspace U ⊂ V that is dominated by N:

 

The Hahn–Banach theorem asserts that φ can be extended to a linear functional on V that is dominated by N.

To derive this from the M. Riesz extension theorem, define a convex cone K ⊂ R×V by

 

Define a functional φ1 on R×U by

 

One can see that φ1 is K-positive, and that K + (R × U) = R × V. Therefore φ1 can be extended to a K-positive functional ψ1 on R×V. Then

 

is the desired extension of φ. Indeed, if ψ(x) > N(x), we have: (N(x), x) ∈ K, whereas

 

leading to a contradiction.

References

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Sources

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  • Castillo, Reńe E. (2005), "A note on Krein's theorem" (PDF), Lecturas Matematicas, 26, archived from the original (PDF) on 2014-02-01, retrieved 2014-01-18
  • Riesz, M. (1923), "Sur le problème des moments. III.", Arkiv för Matematik, Astronomi och Fysik (in French), 17 (16), JFM 49.0195.01
  • Akhiezer, N.I. (1965), The classical moment problem and some related questions in analysis, New York: Hafner Publishing Co., MR 0184042