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

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:

\phi (x)\geq 0\quad {\text{for}}\quad x\in F\cap 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:

\psi |_{F}=\phi \quad {\text{and}}\quad \psi (x)\geq 0\quad {\text{for}}\quad x\in K.

In general, a K-positive linear functional on F cannot be extended to a $K$ -positive linear functional on E. Already in two dimensions one obtains a counterexample taking K to be the upper half plane 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

The proof is similar to the proof of the Hanh-Banach theorem (see also below).

By transfinite induction or Zorn's lemma it is sufficient to consider the case dim E/F = 1.

Choose any y ∈ E\F. Set

$a=\sup\{\phi (x):x\in F,\ y-x\in K\},\ b=\inf\{\phi (x):x\in F,x-y\in K\}.$

We will prove below that -∞ < a ≤ b. For now, choose any c satisfying a ≤ c ≤ b, and set ψ(y) = c, ψ|_F = φ, and then extend ψ to all of E by linearity. We need to show that ψ is K-positive. Suppose z ∈ K. Then either z = 0, or z = p(x + y) or z = p(x - y) for some p > 0 and x ∈ F. If z = 0, then ψ(z) ≥ 0. In the first remaining case x + y = y - (-x) ∈ K, and so

$\psi (y)=c\geq a\geq \phi (-x)=\psi (-x)$

by definition. Thus

$\psi (z)=p\psi (x+y)=p(\psi (x)+\psi (y))\geq 0.$

In the second case, x - y ∈ K, and so similarly

$\psi (y)=c\leq b\leq \phi (x)=\psi (x)$

by definition and so

$\psi (z)=p\psi (x-y)=p(\psi (x)-\psi (y))\geq 0.$

In all cases, ψ(z) ≥ 0, and so ψ is K-positive.

We now prove that -∞ < a ≤ b. Notice by assumption there exists at least one x ∈ F for which y - x ∈ K, and so -∞ <a. However, it may be the case that there are no x ∈ F for which x - y∈ K, in which case b = ∞ and the inequality is trivial (in this case notice that the third case above cannot happen). Therefore, we may assume that b < ∞ and there is at least one x ∈ F for which x - y∈ K. To prove the inequality, it suffices to show that whenever x ∈ F and y - x ∈ K, and x' ∈ F and x' - y ∈ K, then φ(x) ≤ φ(x'). Indeed,

$x'-x=(x'-y)+(y-x)\in K$

since K is a convex cone, and so

$0\leq \phi (x'-x)=\phi (x')-\phi (x)$

since φ is K-positive.

Corollary: Krein's extension theorem

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

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:

\phi (x)\leq N(x),\quad x\in U.

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

K=\left\{(a,x)\,\mid \,N(x)\leq a\right\}.

Define a functional φ₁ on R×U by

\phi _{1}(a,x)=a-\phi (x).

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

\psi (x)=-\psi _{1}(0,x)

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

\psi _{1}(N(x),x)=N(x)-\psi (x)<0,

leading to a contradiction.

References

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

[1] Riesz (1923)

[2] Akhiezer (1965)

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