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M. Riesz extension theorem

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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