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Riesz Representation Theorem


There are a couple of versions of this theorem. Basically, it says that any bounded linear functional T on the space of compactly supported continuous functions on X is the same as integration against a measure mu,

 Tf=intfdmu.

Here, the integral is the Lebesgue integral.

Because linear functionals form a vector space, and are not "positive," the measure mu may not be a positive measure. But if the functional T is positive, in the sense that f>=0 implies that Tf>=0, then the measure mu is also positive. In the generality of complex linear functionals, the measure mu is a complex measure. The measure mu is uniquely determined by T and has the properties of a regular Borel measure. It must be a finite measure, which corresponds to the boundedness condition on the functional. In fact, the operator norm of T, ||T||, is the total variation measure of X, |mu|(X).

Naturally, there are some hypotheses necessary for this to make sense. The space X has to be locally compact and a T2-Space, which is not a strong restriction. In fact, for unbounded spaces X, the theorem also applies to functionals on continuous functions which vanish at infinity, in the sense that for any epsilon>0, there is a compact set K such that for any x not in K, |f(x)|<epsilon (which is the notion from calculus of lim_(x->infty)f(x)=0).

The Riesz representation theorem is useful in describing the dual vector space to any space which contains the compactly supported continuous functions as a dense subspace. Roughly speaking, a linear functional is modified, usually by convolving with a bump function, to a bounded linear functional on the compactly supported continuous functions. Then it can be realized as integration against a measure. Often the measure must be absolutely continuous, and so the dual is integration against a function.


See also

Absolutely Continuous, Complex Measure, Dual Vector Space, Functional, Hilbert Space, Lebesgue Measure, Measure Space, Polar Representation, Radon-Nikodym Theorem, Singular Measure

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Riesz Representation Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/RieszRepresentationTheorem.html

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