Riesz representation theorem of bounded sesquilinear forms
Bounded sesquilinear forms
Let , be two Hilbert spaces.
for all and .
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
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|
|Date of creation||2013-03-22 18:41:38|
|Last modified on||2013-03-22 18:41:38|
|Last modified by||asteroid (17536)|
|Synonym||Riesz lemma on bounded sesquilinear forms|
|Synonym||correspondence between bounded operators and bounded sesquilinear forms|
|Defines||bounded sesquilinear form|