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. The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz during his study of the problem of moments. 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 K {displaystyle 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 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. By transfinite induction it is sufficient to consider the case dim E/F = 1. Choose y ∈ EF. Set and extend ψ to E by linearity. Let us show that ψ is K-positive.