The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz[1] during his study of the problem of moments.
be a real vector space,
be a vector subspace, and
be a convex cone.
A linear functional
-positive, if it takes only non-negative values on the cone
: A linear functional
, and also returns a value of at least 0 for all points in the cone
: In general, a
-positive linear functional on
-positive linear functional on
Already in two dimensions one obtains a counterexample.
The positive functional
can not be extended to a positive functional on
However, the extension exists under the additional assumption that
The proof is similar to the proof of the Hahn–Banach theorem (see also below).
By transfinite induction or Zorn's lemma it is sufficient to consider the case dim
Set We will prove below that
In the first remaining case
, and so similarly by definition and so In all cases,
Notice by assumption there exists at least one
and the inequality is trivial (in this case notice that the third case above cannot happen).
To prove the inequality, it suffices to show that whenever
is a convex cone, and so since
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.
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 φ1 on R×U by 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 is the desired extension of φ.
Indeed, if ψ(x) > N(x), we have: (N(x), x) ∈ K, whereas leading to a contradiction.