M. Riesz extension theorem

M. Riesz extension theorem For more theorems that are sometimes called Riesz's theorem, see Riesz theorem.

The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz[1] during his study of the problem of moments.[2] Contents 1 Formulation 2 Proof 3 Corollary: Krein's extension theorem 4 Connection to the Hahn–Banach theorem 5 Notes 6 References Formulation Let {displaystyle E} be a real vector space, {displaystyle Fsubset E} be a vector subspace, and {displaystyle Ksubset E} be a convex cone.

A linear functional {displaystyle phi :Fto mathbb {R} } is called {displaystyle K} -positive, if it takes only non-negative values on the cone {displaystyle K} : {displaystyle phi (x)geq 0quad {text{for}}quad xin Fcap K.} A linear functional {displaystyle psi :Eto mathbb {R} } is called a {displaystyle K} -positive extension of {displaystyle phi } , if it is identical to {displaystyle phi } in the domain of {displaystyle phi } , and also returns a value of at least 0 for all points in the cone {displaystyle K} : {displaystyle psi |_{F}=phi quad {text{and}}quad psi (x)geq 0quad {text{for}}quad xin K.} In general, a {displaystyle K} -positive linear functional on {displaystyle F} cannot be extended to a {displaystyle K} -positive linear functional on {displaystyle E} . Already in two dimensions one obtains a counterexample. Let {displaystyle E=mathbb {R} ^{2}, K={(x,y):y>0}cup {(x,0):x>0},} and {displaystyle F} be the {displaystyle x} -axis. The positive functional {displaystyle phi (x,0)=x} can not be extended to a positive functional on {displaystyle E} .

However, the extension exists under the additional assumption that {displaystyle Esubset K+F,} namely for every {displaystyle yin E,} there exists an {displaystyle xin F} such that {displaystyle y-xin K.} Proof 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  {displaystyle E/F=1} .

Choose any {displaystyle yin Esetminus F} . Set {displaystyle a=sup{,phi (x)mid xin F, y-xin K,}, b=inf{,phi (x)mid xin F,x-yin K,}.} We will prove below that {displaystyle -infty 0} and {displaystyle xin F} . If {displaystyle z=0} , then {displaystyle psi (z)>0} . In the first remaining case {displaystyle x+y=y-(-x)in K} , and so {displaystyle psi (y)=cgeq ageq phi (-x)=psi (-x)} by definition. Thus {displaystyle psi (z)=ppsi (x+y)=p(psi (x)+psi (y))geq 0.} In the second case, {displaystyle x-yin K} , and so similarly {displaystyle psi (y)=cleq bleq phi (x)=psi (x)} by definition and so {displaystyle psi (z)=ppsi (x-y)=p(psi (x)-psi (y))geq 0.} In all cases, {displaystyle psi (z)>0} , and so {displaystyle psi } is {displaystyle K} -positive.

We now prove that {displaystyle -infty  0.

Connection to the Hahn–Banach theorem Main article: Hahn–Banach theorem 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: {displaystyle phi (x)leq N(x),quad xin U.} 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 {displaystyle K=left{(a,x),mid ,N(x)leq aright}.} Define a functional φ1 on R×U by {displaystyle phi _{1}(a,x)=a-phi (x).} 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 {displaystyle psi (x)=-psi _{1}(0,x)} is the desired extension of φ. Indeed, if ψ(x) > N(x), we have: (N(x), x) ∈ K, whereas {displaystyle psi _{1}(N(x),x)=N(x)-psi (x)<0,} leading to a contradiction. Notes ^ Riesz (1923) ^ Akhiezer (1965) References Castillo, Reńe E. (2005), "A note on Krein's theorem" (PDF), Lecturas Matematicas, 26, archived from the original (PDF) on 2014-02-01, retrieved 2014-01-18 Riesz, M. (1923), "Sur le problème des moments. III.", Arkiv för Matematik, Astronomi och Fysik (in French), 17 (16), JFM 49.0195.01 Akhiezer, N.I. (1965), The classical moment problem and some related questions in analysis, New York: Hafner Publishing Co., MR 0184042 hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Theorems in convex geometryTheorems in functional analysis