Riesz representation theorem

There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.

The Hilbert space representation theorem

This theorem establishes an important connection between a Hilbert space and its (continuous) dual space: if the underlying field is the real numbers, the two are isometrically isomorphic; if the field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural one as will be described next.

Let $H$ be a Hilbert space, and let $H^*$ denote its dual space, consisting of all continuous linear functionals from $H$ into the field $mathbb\{R\}$ or $mathbb\{C\}$. If $x$ is an element of $H$, then the function $phi\_x$ defined by

:$phi\_x(y)\; =\; leftlangle\; y\; ,\; x\; ight\; angle\; quad\; forall\; y\; in\; H$

where $langlecdot,cdot\; angle$ denotes the inner product of the Hilbert space, is an element of $H^*$. The Riesz representation theorem states that every element of $H^*$ can be written uniquely in this form.

Theorem. The mapping

:$Phi:H\; ightarrow\; H^*,\; quad\; Phi(x)\; =\; phi\_x$

is an isometric (anti-) isomorphism, meaning that:

* $Phi$ is bijective.
* The norms of $x$ and $Phi(x)$ agree: $Vert\; x\; Vert\; =\; VertPhi(x)Vert$.
* $Phi$ is additive: $Phi(\; x\_1\; +\; x\_2\; )\; =\; Phi(\; x\_1\; )\; +\; Phi(\; x\_2\; )$.
* If the base field is $mathbb\{R\}$, then $Phi(lambda\; x)\; =\; lambda\; Phi(x)$ for all real numbers $lambda$.
* If the base field is $mathbb\{C\}$, then for all complex numbers $lambda$, where denotes the complex conjugation of $lambda$.

The inverse map of $Phi$ can be described as follows. Given an element $phi$ of $H^*$, the orthogonal complement of the kernel of $phi$ is a one-dimensional subspace of $H$. Take a non-zero element $z$ in that subspace, and set $x\; =\; phi(z)\; cdot\; z\; /\{leftVert\; z\; ightVert\}^2$. Then Φ("x") = φ.

Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907 (see references). Gray (1984) starts his review on the development up to the Riesz representation theorem with what he considers the pristine form in Riesz (1909): "Given the operation $A\; [f]$, one can construct the function of bounded variation $alpha(x)$, such that, whatever the continuous function $f(x)$ is, one has $A\; [f]\; =\; int\_\{0\}^\{1\}\; f(x),dalpha(x).$"

In the mathematical treatment of quantum mechanics, the theorem can be seen as a justification for the popular bra-ket notation. When the theorem holds, every ket $|psi\; angle$ has a corresponding bra $langlepsi|$, and the correspondence is unambiguous. However, there are topological vector spaces, such as nuclear spaces, where the Riesz repesentation theorem does not hold, in which case the bra-ket notation can become awkward.

The representation theorem for linear functionals on C_c("X")

The following theorem represents positive linear functionals on C_c("X"), the space of continuous complex valued functions of compact support. The Borel sets in the following statement refers to the σ-algebra generated by the "open" sets.

A non-negative countably additive Borel measure μ on a locally compact Hausdorff space "X" is regular if and only if

* &mu;("K") < &infin; for every compact "K";

* For every Borel set "E",:$mu(E)\; =\; inf\; \{mu(U):\; E\; subseteq\; U,\; U\; mbox\{\; open\}\}$

* The relation:$mu(E)\; =\; sup\; \{mu(K):\; K\; subseteq\; E,\; K\; mbox\{\; compact\}\}$holds whenever "E" is open or when "E" is Borel and &mu;(E) < &infin;.

Theorem. Let "X" be a locally compact Hausdorff space. For any positive linear functional &psi; on C_c("X"), there is a unique Borel regular measure &mu; on "X" such that:$psi(f)\; =\; int\_X\; f(x)\; ,\; d\; mu(x)\; quad$for all "f" in C_c("X").

One approach to measure theory is to start with a Radon measure, defined as a positive linear functional on "C(X)". This is the way adopted by Bourbaki; it does of course assume that "X" starts life as a topological space, rather than simply as a set. For locally compact spaces an integration theory is then recovered.

The representation theorem for the dual of C₀("X")

The following theorem, also referred to as the "Riesz-Markov theorem" gives a concrete realisation of the dual space of C₀("X"), the set of continuous functions on "X" which vanish at infinity. The Borel sets in the statement of the theorem also refers to the &sigma;-algebra generated by the "open" sets. This result is similar to the result of the preceding section, but it does not subsume the previous result. See the technical remark below.

If &mu; is a complex-valued countably additive Borel measure, &mu; is regular if and only if the non-negative countably additive measure |&mu;| is regular as defined above.

Theorem. Let "X" be a locally compact Hausdorff space. For any continuous linear functional &psi; on C₀("X"), there is a unique "regular" countably additive complex Borel measure &mu; on "X" such that:$psi(f)\; =\; int\_X\; f(x)\; ,\; d\; mu(x)\; quad$for all "f" in C₀("X"). The norm of &psi; as a linear functional is the total variation of &mu;, that is:$|psi|\; =\; |mu|(X).$Finally, &psi; is positive if and only if the measure &mu; is non-negative.

Remark. Every bounded linear functional on C_c("X") extends uniquely to a bounded linear functional on C₀("X") since the latter space is the closure of the former. However, an unbounded positive linear functional on C_c("X") does not extend to a "bounded" linear functional on C₀("X"). For this reason the previous results apply to slightly different situations.

References

* M. Fréchet (1907). Sur les ensembles de fonctions et les opérations linéaires. "C. R. Acad. Sci. Paris" 144, 1414&ndash;1416.
* F. Riesz (1907). Sur une espèce de géométrie analytiques des systèmes de fonctions sommables. "C. R. Acad. Sci. Paris" 144, 1409&ndash;1411.
* F. Riesz (1909). Sur les opérations fonctionelles linéaires. "C. R. Acad. Sci. Paris" "149", 974&ndash;977.
* J. D. Gray, The shaping of the Riesz representation theorem: A chapter in the history of analysis, Archive for History in the Exact Sciences, Vol 31(3) 1984-85, 127-187.
* P. Halmos "Measure Theory", D. van Nostrand and Co., 1950.
* P. Halmos, "A Hilbert Space Problem Book", Springer, New York 1982 "(problem 3 contains version for vector spaces with coordinate systems)".
* D. G. Hartig, The Riesz representation theorem revisited, "American Mathematical Monthly", 90(4), 277-280 "(A category theoretic presentation as natural transformation)".
* Walter Rudin, "Real and Complex Analysis", McGraw-Hill, 1966, ISBN 0-07-100276-6.

*
*
* [http://nfist.ist.utl.pt/~edgarc/wiki/index.php/Riesz_representation_theorem Proof of Riesz representation theorem in Hilbert spaces] on [http://bourbawiki.no-ip.org Bourbawiki]