bounded linear functionals on
For any measure space and , the following linear map can be defined
It is easily shown that is bounded (http://planetmath.org/OperatorNorm), so is a member of the dual space of . However, unless the measure space consists of a finite set of atoms, not every element of the dual of can be written like this. Instead, it is necessary to restrict to linear maps satisfying a bounded convergence property.
Let be a -finite (http://planetmath.org/SigmaFinite) measure space and be the space of bounded linear maps satisfying bounded convergence. That is, if are in and for almost every , then .
Then gives an isometric isomorphism from to .
First, the operator norm is equal to the -norm of (see -norm is dual to (http://planetmath.org/LpNormIsDualToLq)), so the map gives an isometric embedding from into the dual of . Furthermore, dominated convergence implies that satisfies bounded convergence so . We just need to show that maps onto .
So, suppose that . It needs to be shows that for some . Defining an additive set function (http://planetmath.org/Additive) by
for every set , the bounded convergence property for implies that is countably additive and is therefore a finite signed measure. So, the Radon-Nikodym theorem gives a such that for every . Then, the equality
is satisfied for with any and the functional monotone class theorem extends this to any bounded and measurable , giving . ∎
|Title||bounded linear functionals on|
|Date of creation||2013-03-22 18:38:08|
|Last modified on||2013-03-22 18:38:08|
|Last modified by||gel (22282)|