proof of -norm is dual to
Furthermore, if either and or then is not required to be -finite.
If then is zero almost everywhere, and both sides of equality (1) are zero. So, we only need to consider the case where .
If and then, setting gives
On the other hand, if so that , then setting gives and
Now consider the case where and . Let be the sequence of functions
then, and monotone convergence gives . Therefore,
and letting go to infinity gives .
We finally consider . Then, for any there exists a set with such that on . Also, monotone convergence gives and, therefore, eventually. Replacing by if necessary, we may suppose that . So, setting gives and,
Letting increase to gives as required.
|Title||proof of -norm is dual to|
|Date of creation||2013-03-22 18:38:16|
|Last modified on||2013-03-22 18:38:16|
|Last modified by||gel (22282)|