# 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]}

## Contents

## Formulation[edit]

Let *E* be a real vector space, *F* ⊂ *E* a vector subspace, and let *K* ⊂ *E* be a convex cone.

A linear functional *φ*: *F* → **R** is called *K*-*positive*, if it takes only non-negative values on the cone *K*:

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

In general, a *K*-positive linear functional on *F* cannot be extended to a -positive linear functional on *E*. Already in two dimensions one obtains a counterexample taking *K* to be the upper halfplane with the open negative *x*-axis removed. If *F* is the *x*-axis, then the positive functional *φ*(*x*, 0) = *x* can not be extended to a positive functional on the plane.

However, the extension exists under the additional assumption that for every *y* ∈ *E* there exists *x*∈*F* such that *y* − *x* ∈*K*; in other words, if *E* = *K* + *F*.

## Proof[edit]

By transfinite induction it is sufficient to consider the case dim *E*/*F* = 1.

Choose *y* ∈ *E*\*F*. Set

and extend *ψ* to *E* by linearity. Let us show that *ψ* is *K*-positive.

Every point *z* in *K* is a positive linear multiple of either *x* + *y* or *x* − *y* for some *x* ∈ *F*. In the first case, *z* = *a*(*y* + *x*), therefore *y*− *(*−*x)* = *z*/*a* is in *K* with −*x* in *F* . Hence

therefore *ψ*(*z*) ≥ 0. In the second case, *z* = *a*(*x* − *y*), therefore *y* = *x* − *z*/*a*. Let *x*_{1} ∈ *F* be such that *z*_{1} = *y* − *x*_{1} ∈ *K* and *ψ*(*x*_{1}) ≥ ψ(*y*) − *ε*. Then

therefore *ψ*(*z*) ≥ −*a* *ε*. Since this is true for arbitrary *ε* > 0, we obtain *ψ*(*z*) ≥ 0.

## Corollary: Krein's extension theorem[edit]

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 *φ*: *E* → **R** such that *φ*(*x*) > 0.

## Connection to the Hahn–Banach theorem[edit]

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 *φ*_{1} on **R**×*U* by

One can see that *φ*_{1} is *K*-positive, and that *K* + (**R** × *U*) = **R** × *V*. Therefore *φ*_{1} can be extended to a *K*-positive functional *ψ*_{1} 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.

## Notes[edit]

## References[edit]

- Castillo, Reńe E. (2005), "A note on Krein's theorem" (PDF),
*Lecturas Matematicas*,**26** - 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