Riesz representation theorem
There are several well-known theorems in
functional analysisknown 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 spaceand 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 be a Hilbert space, and let denote its dual space, consisting of all
continuous linear functionals from into the field or . If is an element of , then the function defined by
where denotes the
inner productof the Hilbert space, is an element of . The Riesz representation theorem states that every element of can be written uniquely in this form.
Theorem. The mapping
is an isometric (anti-) isomorphism, meaning that:
* The norms of and agree: .
* is additive: .
* If the base field is , then for all real numbers .
* If the base field is , then for all complex numbers , where denotes the complex conjugation of .
The inverse map of can be described as follows. Given an element of , the orthogonal complement of the kernel of is a one-dimensional subspace of . Take a non-zero element in that subspace, and set . 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 , one can construct the function of
bounded variation, such that, whatever the continuous function is, one has "
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 has a corresponding bra , 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 Cc("X")
The following theorem represents positive linear functionals on Cc("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
* μ("K") < ∞ for every compact "K";
* For every Borel set "E",:
* The relation:holds whenever "E" is open or when "E" is Borel and μ(E) < ∞.
Theorem. Let "X" be a
locally compact Hausdorff space. For any positive linear functionalψ on Cc("X"), there is a unique Borel regular measureμ on "X" such that:for all "f" in Cc("X").
One approach to
measure theoryis 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 spaceof 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 σ-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 μ is a complex-valued countably additive Borel measure, μ is regular if and only if the non-negative countably additive measure |μ| is regular as defined above.
Theorem. Let "X" be a
locally compact Hausdorff space. For any continuous linear functionalψ on C0("X"), there is a unique "regular" countably additive complex Borel measure μ on "X" such that:for all "f" in C0("X"). The norm of ψ as a linear functional is the total variation of μ, that is:Finally, ψ is positive if and only if the measure μ is non-negative.
Remark. Every bounded linear functional on Cc("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 Cc("X") does not extend to a "bounded" linear functional on C0("X"). For this reason the previous results apply to slightly different situations.
* M. Fréchet (1907). Sur les ensembles de fonctions et les opérations linéaires. "C. R. Acad. Sci. Paris" 144, 1414–1416.
* F. Riesz (1907). Sur une espèce de géométrie analytiques des systèmes de fonctions sommables. "C. R. Acad. Sci. Paris" 144, 1409–1411.
* F. Riesz (1909). Sur les opérations fonctionelles linéaires. "C. R. Acad. Sci. Paris" "149", 974–977.
* 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]
Wikimedia Foundation. 2010.
Look at other dictionaries:
Representation theorem — In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to a concrete structure. For example, *in algebra, ** Cayley s theorem states that every group is isomorphic to… … Wikipedia
Riesz theorem — See:* F. and M. Riesz theorem * Riesz representation theorem * M. Riesz extension theorem * Riesz Thorin theorem * Riesz Fischer theoremFrigyes Riesz and Marcel Riesz were two brothers, both of whom were notable mathematicians … Wikipedia
Frigyes Riesz — Infobox Scientist name = Frigyes Riesz box width = image width = caption = birth date = 1880 01 22 birth place = Győr, Hungary (Austria Hungary) death date = death date and age|1956|2|28|1880|1|22 death place = Budapest, Hungary residence =… … Wikipedia
Théorème de Riesz — Cette page d’homonymie répertorie les différents sujets et articles partageant un même nom. Plusieurs noms de théorèmes font référence aux deux frères Riesz, mathématiciens hongrois : Frigyes Riesz Théorème de compacité de Riesz, qui dit qu… … Wikipédia en Français
Hilbert space — For the Hilbert space filling curve, see Hilbert curve. Hilbert spaces can be used to study the harmonics of vibrating strings. The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It… … Wikipedia
List of mathematics articles (R) — NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… … Wikipedia
List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… … Wikipedia
Vector space — This article is about linear (vector) spaces. For the structure in incidence geometry, see Linear space (geometry). Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is… … Wikipedia
Self-adjoint operator — In mathematics, on a finite dimensional inner product space, a self adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose.… … Wikipedia
Spectral theory of ordinary differential equations — In mathematics, the spectral theory of ordinary differential equations is concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl… … Wikipedia