Riesz representation theorem (of linear functionals on function spaces)
This entry should not be mistaken with the entry on the Riesz representation theorem^{} of bounded^{} (http://planetmath.org/BoundedOperator) linear functionals^{} on an Hilbert space^{}.
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
$$f\u27fc\int f\mathit{d}\mu $$ 
for some measure $\mu $.
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:

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$X$ is a locally compact Hausdorff space^{}.

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${C}_{c}(X)$ denotes the space of real valued continuous functions^{} on $X$ with compact support.

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${C}_{0}(X)$ denotes the space of real valued continuous functions on $X$ that vanish at infinity.

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all function spaces are endowed with the supnorm $\parallel .{\parallel}_{\mathrm{\infty}}$

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a linear functional $L$ is said to be if $0\le L(f)$ whenever $0\le f$.
Theorem 1 (RieszMarkov)  Let $L$ be a positive linear functional^{} on ${C}_{c}(X)$. There exists a unique Radon measure^{} $\mu $ on $X$, whose underlying $\sigma $algebra (http://planetmath.org/SigmaAlgebra) is the $\sigma $algebra generated by all compact sets, such that
$$L(f)={\int}_{X}f\mathit{d}\mu $$ 
Moreover, $\mu $ is finite if and only if $L$ is bounded.
Notice that when $X$ is $\sigma $compact (http://planetmath.org/SigmaCompact) the underlying $\sigma $algebra for these measures is precisely the Borel $\sigma $algebra (http://planetmath.org/BorelSigmaAlgebra) of $X$.
$$
Theorem 2  Let $L$ be a positive linear functional on ${C}_{0}(X)$. There exists a unique finite Radon measure $\mu $ on $X$ such that
$$L(f)={\int}_{X}f\mathit{d}\mu $$ 
Theorem 3 (Dual of ${C}_{\mathrm{0}}\mathbf{}\mathrm{(}X\mathrm{)}$)  Let $L$ be a linear functional on ${C}_{0}(X)$. There exists a unique finite signed (http://planetmath.org/SignedMeasure) Borel measure on $X$ such that
$$L(f)={\int}_{X}f\mathit{d}\mu $$ 
0.0.1 Complex version:
Here ${C}_{0}(X)$ denotes the space of complex valued continuous functions on $X$ that vanish at infinity.
Theorem 4  Let $L$ be a linear functional on ${C}_{0}(X)$. There exists a unique finite complex Borel measure $\mu $ on $X$ such that
$$L(f)={\int}_{X}f\mathit{d}\mu $$ 
Title  Riesz representation theorem (of linear functionals on function spaces) 

Canonical name  RieszRepresentationTheoremofLinearFunctionalsOnFunctionSpaces 
Date of creation  20130322 17:28:18 
Last modified on  20130322 17:28:18 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  13 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 28C05 
Classification  msc 46A99 
Defines  RieszMarkov theorem 