Riesz representation theorem of bounded sesquilinear forms


Bounded sesquilinear forms

Let H1, H2 be two Hilbert spacesMathworldPlanetmath.

Definition - A sesquilinear formPlanetmathPlanetmath [,]:H1×H2 is said to be boundedPlanetmathPlanetmath if there is a constant C0 such that

[ξ,η]Cξη

for all ξH1 and ηH2.

Bounded sesquilinear forms are precisely those which are continuous from H1×H2 to .

Examples :

  • When H1 and H2 are the same Hilbert space, denoted by H, the inner productMathworldPlanetmath , in H is itself a bounded sesquilinear form. The boundedness condition follows from the Cauchy-Schwarz inequality.

  • Let T:H1H2 be a bounded linear operator and denote by , the inner product in H2. The function [,]:H1×H2 defined by

    [ξ,η]:=Tξ,η

    is a bounded sesquilinear form. The boundedness condition follows from the Cauchy-Schwarz inequality and the fact that T is bounded.

Riesz representation of bounded sesquilinear forms

The second example above is in fact the general case. To every bounded sesquilinear form one can associate to it a unique bounded operatorMathworldPlanetmathPlanetmath. That is content of the following result:

Theorem - Riesz - Let H1, H2 be two Hilbert spaces and denote by , the inner product in H2. For every bounded sesquilinear form [,]:H1×H2 there is a unique bounded linear operator T:H1H2 such that

[ξ,η]=Tξ,η,ξH1,ηH2.

Thus, there is a correspondence between bounded linear operators and bounded sesquilinear forms. Actually, in the early twentieth century, spectral theory was formulated solely in terms of sesquilinear forms on Hilbert spaces. Only later it was realized that this could be achieved, perhaps in a more intuitive manner, by considering linear operatorsMathworldPlanetmath instead. The linear operator approach has its advantages, as for example one can define the composition of linear operators but not of sesquilinear forms. Nevertheless it is many times useful to define a linear operator by specifying its sesquilinear form.

Title Riesz representation theorem of bounded sesquilinear forms
Canonical name RieszRepresentationTheoremOfBoundedSesquilinearForms
Date of creation 2013-03-22 18:41:38
Last modified on 2013-03-22 18:41:38
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 10
Author asteroid (17536)
Entry type Theorem
Classification msc 47A07
Classification msc 46C05
Synonym Riesz lemma on bounded sesquilinear forms
Synonym correspondence between bounded operators and bounded sesquilinear forms
Defines bounded sesquilinear form