proof of basic criterion for self-adjointness
If holds, then , so that is dense in . Also, since is symmetric, for ,
because . Hence , so that given a sequence such that , we have that is a Cauchy sequence and thus itself is a Cauchy sequence. Hence converges to some and since is closed it follows that and . This proves that , so that is closed (and similarly, is closed. Thus .
Suppose . If , then there is such that . Since is symmetric, , so that . But since , it follows that , so that . Hence , and therefore is self-adjoint.
|Title||proof of basic criterion for self-adjointness|
|Date of creation||2013-03-22 14:53:05|
|Last modified on||2013-03-22 14:53:05|
|Last modified by||Koro (127)|