proof of bounded linear functionals on
For any , define the linear map
This is a bounded linear map with operator norm (see -norm is dual to (http://planetmath.org/LpNormIsDualToLq)), so the map gives an isometric embedding from to the dual space of . It only remains to show that it is onto.
So, suppose that is a bounded linear map. It needs to be shown that there is a with . As any -finite measure is equivalent to a probability measure (http://planetmath.org/AnySigmaFiniteMeasureIsEquivalentToAProbabilityMeasure), there is a bounded such that . Let be the bounded linear map given by . Then, there is a such that
Letting tend to infinity, dominated convergence says that in the -norm, so Fatou’s lemma gives
In particular, (see -norm is dual to (http://planetmath.org/LpNormIsDualToLq)), so . As are in , dominated convergence finally gives
so as required.
|Title||proof of bounded linear functionals on|
|Date of creation||2013-03-22 18:38:19|
|Last modified on||2013-03-22 18:38:19|
|Last modified by||gel (22282)|