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There are several well-known theorems in [[functional analysis]] known as the '''Riesz representation theorem'''. They are named in honour of [[Frigyes Riesz]].
 
This article will describe his theorem concerning the dual of a [[Hilbert space]], which is sometimes called the Fréchet-Riesz theorem. For the theorems relating linear functionals to measures, see [[Riesz–Markov–Kakutani representation theorem]].
 
== The Hilbert space representation theorem ==
This theorem establishes an important connection between a [[Hilbert space]] and its (continuous) [[continuous dual space|dual space]]. If the underlying [[Field (mathematics)|field]] is the [[real number]]s, the two are [[isometry|isometrically]] [[isomorphism|isomorphic]]; if the underlying field is the [[complex number]]s, the two are isometrically [[antiisomorphic|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 functional]]s from ''H'' into the field '''R''' or '''C'''. If ''x'' is an element of ''H'', then the function φ<sub>''x''</sub>, defined by
 
:<math>\phi_x (y) = \left\langle y , x \right\rangle \quad \forall y \in H </math>
 
where <math>\langle\cdot,\cdot\rangle</math> 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 Φ: ''H'' → ''H*'' defined by Φ(''x'') = φ<sub>''x''</sub> is an isometric (anti-) isomorphism, meaning that:
* Φ is [[bijective]].
* The norms of ''x'' and Φ(''x'') agree: <math>\Vert x \Vert = \Vert\Phi(x)\Vert</math>.
* Φ is additive: <math>\Phi( x_1 + x_2 ) = \Phi( x_1 ) + \Phi( x_2 )</math>.
* If the base field is '''R''', then <math>\Phi(\lambda x) = \lambda \Phi(x)</math> for all real numbers λ.
* If the base field is '''C''', then <math>\Phi(\lambda x) = \bar{\lambda} \Phi(x)</math> for all complex numbers λ, where <math>\bar{\lambda}</math> denotes the complex conjugation of λ.
 
The inverse map of Φ can be described as follows. Given an element φ of ''H*'', the orthogonal complement of the kernel of φ is a one-dimensional subspace of ''H''. Take a non-zero element ''z'' in that subspace, and set <math>x = \overline{\varphi(z)} \cdot z /{\left\Vert z \right\Vert}^2</math>. Then Φ(''x'') = φ.
 
Historically, the theorem is often attributed simultaneously to [[Frigyes Riesz|Riesz]] and [[Maurice René Fréchet|Fréchet]] in 1907 (see references).
 
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 <math>|\psi\rangle</math> has a corresponding bra <math>\langle\psi|</math>, and the correspondence is unambiguous.
 
== References ==
* M. Fréchet (1907). Sur les ensembles de fonctions et les opérations linéaires. ''[[Les Comptes rendus de l'Académie des sciences|C. R. Acad. Sci. Paris]]'' '''144''', 1414&ndash;1416.
* F. Riesz (1907). Sur une espèce de géométrie analytique des systèmes de fonctions sommables. ''C. R. Acad. Sci. Paris'' '''144''', 1409&ndash;1411.
* F. Riesz (1909). Sur les opérations fonctionnelles 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(2) 1984&ndash;85, 127&ndash;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&ndash;280 ''(A category theoretic presentation as natural transformation)''.
* {{springer|title=Riesz representation theorem|id=p/r027172}}
* Walter Rudin, ''Real and Complex Analysis'', McGraw-Hill, 1966, ISBN 0-07-100276-6.
* {{mathworld|urlname=RieszRepresentationTheorem|title=Riesz Representation Theorem}}
* {{planetmath reference|id=6130|title=Proof of Riesz representation theorem for separable Hilbert spaces}}
 
[[Category:Theorems in functional analysis]]
[[Category:Duality theories]]
[[Category:Integral representations]]

Revision as of 05:35, 29 January 2014

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

This article will describe his theorem concerning the dual of a Hilbert space, which is sometimes called the Fréchet-Riesz theorem. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.

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 underlying 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 R or C. If x is an element of H, then the function φx, defined by

ϕx(y)=y,xyH

where , 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 Φ: HH* defined by Φ(x) = φx is an isometric (anti-) isomorphism, meaning that:

The inverse map of Φ can be described as follows. Given an element φ of H*, the orthogonal complement of the kernel of φ is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set x=φ(z)z/z2. Then Φ(x) = φ.

Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907 (see references).

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 |ψ has a corresponding bra ψ|, and the correspondence is unambiguous.

References

  • 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 analytique des systèmes de fonctions sommables. C. R. Acad. Sci. Paris 144, 1409–1411.
  • F. Riesz (1909). Sur les opérations fonctionnelles 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(2) 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).
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  • Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.
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