Riesz representation theorem of bounded sesquilinear forms
Bounded sesquilinear forms
Let , be two Hilbert spaces.
Definition - A sesquilinear form is said to be bounded if there is a constant such that
for all and .
Bounded sesquilinear forms are precisely those which are continuous from to .
Examples :
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When and are the same Hilbert space, denoted by , the inner product in is itself a bounded sesquilinear form. The boundedness condition follows from the Cauchy-Schwarz inequality.
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Let be a bounded linear operator and denote by the inner product in . The function defined by
is a bounded sesquilinear form. The boundedness condition follows from the Cauchy-Schwarz inequality and the fact that 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 operator. That is content of the following result:
Theorem - Riesz - Let , be two Hilbert spaces and denote by the inner product in . For every bounded sesquilinear form there is a unique bounded linear operator such that
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 operators 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 |
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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 |