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.

Formulation

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

A linear functional $\phi :F\to \mathbb {R}$ is called $K$-positive, if

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

A linear functional $\psi :E\to \mathbb {R}$ is called a $K$-positive extension of $F$ if

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

In general, a $K$-positive linear functional on $F$ can not be extended to a $K$-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 closed positive x-axis, then the positive functional $\phi (x,0)=x$ can not be extended to a positive functional on the plane.

However, the extension exists under the additional assumption that for every $y\in E$ there exist $x,x'\in F$ such that $x-y\in K$ and $y-x'\in K.$

Proof

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

Choose a $y\in E\setminus F$. Let $\psi (y):=\sup _{x'}\phi (x')$, where the supremum is taken over $x'$ satisfying the additional assumption. Extend $\psi$ by linearity to $E$ so as to extend $\phi$. We claim that $\psi$ is $K$-positive.

Every point in $E\setminus F$ is a positive linear multiple of either $x-y$ or $y-x'$ for some $x,x'\in K$, and by linearity it suffices to verify positivity in these two cases.

If $y-x'\in K$, then $\psi (y-x')\geq 0$ by definition of $\psi (y)$.

If $x-y\in K$, then $\psi (x-y)=\phi (x-x')+\psi (x'-y)$ for every $x'$ as in the additional assumption. But the first term is positive since $x-x'=x-y+y-x'\in K$ and the second can be arbitrarily small by definition of $\psi (y)$. Therefore $\psi (x-y)\geq 0$.

Corollary: Krein's extension theorem

Let $E$ be a real linear space, and let $K\subset E$ be a convex cone. Let $x\in E$ be such that for every $y\in E$, there exist $a,b\in \mathbb {R}$ such that $ax-y\in K$ and $bx+y\in K$. If $x\not \in -K$, then there exists a $K$-positive linear functional $\phi :E\to \mathbb {R}$ such that $\phi (x)>0$.

See also

• Hahn–Banach theorem

References

• 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.