M. Riesz extension theorem

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

[edit] Formulation

Let $E$ be a real linear space, $F \subset E$ be a linear subspace, and let $K \subset E$ be a convex cone.

Then, a linear functional $\phi:F\to \mathbb R$ that is $K$ -positive, meaning that

$\phi(x) \geq 0 \quad \text{for} \quad x \in F \cap K$ ,

can be extended to a $K$ -positive linear functional on $E$ . In other words, there exists a linear functional

$\psi:E\to \mathbb R$

such that

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

The proof of this theorem uses transfinite induction. The main step is to show that the theorem holds if $\dim E/F = 1$ .

Let $E$ be a real linear space, and let $K \subset E$ be a convex cone.

For every $0 \neq x \in E$ , there exists a $K$ -positive linear functional $\phi: E \to \mathbb{R}$ such that $\phi(x) \neq 0$ .

M.Riesz, Sur le problème des moments, 1923
N.I.Akhiezer, The classical moment problem and some related questions in analysis, Translated from the Russian by N. Kemmer, Hafner Publishing Co., New York 1965 x+253 pp.