existence of adjoints of bounded operators
Proof : Since is densely defined and bounded, it extends uniquely to a bounded (everywhere defined) linear operator on , which we denote by .
Since extends , we also have that for every there exists such that
We conclude that is everywhere defined. To see that it is bounded one just needs to check that
where the last inequality comes from the Cauchy-Schwarz inequality and the fact that is bounded.
Remark - This theorem shows in particular that bounded linear operators have bounded adjoints .
|Title||existence of adjoints of bounded operators|
|Date of creation||2013-03-22 17:33:44|
|Last modified on||2013-03-22 17:33:44|
|Last modified by||asteroid (17536)|
|Synonym||bounded operators have (bounded) adjoints|