Riesz representation theorem (of linear functionals on function spaces)
The Riesz provided here basically that linear functionals on certain spaces of functions can be seen as integration against measures. In other , for some spaces of functions all linear functionals have the form
for some measure .
There are many versions of these Riesz , and which version is used depends upon the generality wishes to achieve, the difficulty of proof, the of space of functions involved, the of linear functionals involved, the of the ”” space involved, and also the of measures involved.
We present here some possible Riesz of general use.
Notation - In the following we adopt the following conventions:
Theorem 1 (Riesz-Markov) - Let be a positive linear functional on . There exists a unique Radon measure on , whose underlying -algebra (http://planetmath.org/SigmaAlgebra) is the -algebra generated by all compact sets, such that
Moreover, is finite if and only if is bounded.
Notice that when is -compact (http://planetmath.org/SigmaCompact) the underlying -algebra for these measures is precisely the Borel -algebra (http://planetmath.org/BorelSigmaAlgebra) of .
Theorem 2 - Let be a positive linear functional on . There exists a unique finite Radon measure on such that
Theorem 3 (Dual of ) - Let be a linear functional on . There exists a unique finite signed (http://planetmath.org/SignedMeasure) Borel measure on such that
0.0.1 Complex version:
Here denotes the space of complex valued continuous functions on that vanish at infinity.
Theorem 4 - Let be a linear functional on . There exists a unique finite complex Borel measure on such that
|Title||Riesz representation theorem (of linear functionals on function spaces)|
|Date of creation||2013-03-22 17:28:18|
|Last modified on||2013-03-22 17:28:18|
|Last modified by||asteroid (17536)|