M. Riesz extension theorem
- 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 be a real vector space, be a vector subspace, and let be a convex cone.
Then, a linear functional that is -"positive", meaning that
:,
can be extended to a -positive linear functional on . In other words,there exists a linear functional
:
such that
: for .
The proof of this theorem uses transfinite induction. The main step is to show that the theorem holds if .
Corollary: Krein's extension theorem
Let be a real linear space, and let be a convex cone.
For every , there exists a -positive linear functional such that .
ee 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.
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