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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]].
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| 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]].
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| == The Hilbert space representation theorem ==
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| 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.
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| 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
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| :<math>\phi_x (y) = \left\langle y , x \right\rangle \quad \forall y \in H </math> | |
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| 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.
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| '''Theorem'''. The mapping Φ: ''H'' → ''H*'' defined by Φ(''x'') = φ<sub>''x''</sub> is an isometric (anti-) isomorphism, meaning that:
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| * Φ is [[bijective]].
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| * The norms of ''x'' and Φ(''x'') agree: <math>\Vert x \Vert = \Vert\Phi(x)\Vert</math>.
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| * Φ is additive: <math>\Phi( x_1 + x_2 ) = \Phi( x_1 ) + \Phi( x_2 )</math>.
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| * If the base field is '''R''', then <math>\Phi(\lambda x) = \lambda \Phi(x)</math> for all real numbers λ.
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| * 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.
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| 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.
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| Historically, the theorem is often attributed simultaneously to [[Frigyes Riesz|Riesz]] and [[Maurice René Fréchet|Fréchet]] in 1907 (see references).
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| 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.
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| == References ==
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| * 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''', 1414–1416.
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| * F. Riesz (1907). Sur une espèce de géométrie analytique des systèmes de fonctions sommables. ''C. R. Acad. Sci. Paris'' '''144''', 1409–1411.
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| * F. Riesz (1909). Sur les opérations fonctionnelles linéaires. ''C. R. Acad. Sci. Paris'' ''149'', 974–977.
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| * 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.
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| * [[P. Halmos]] ''Measure Theory'', D. van Nostrand and Co., 1950.
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| * P. Halmos, ''A Hilbert Space Problem Book'', Springer, New York 1982 ''(problem 3 contains version for vector spaces with coordinate systems)''.
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| * 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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| * {{springer|title=Riesz representation theorem|id=p/r027172}}
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| * Walter Rudin, ''Real and Complex Analysis'', McGraw-Hill, 1966, ISBN 0-07-100276-6.
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| * {{mathworld|urlname=RieszRepresentationTheorem|title=Riesz Representation Theorem}}
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| * {{planetmath reference|id=6130|title=Proof of Riesz representation theorem for separable Hilbert spaces}}
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| [[Category:Theorems in functional analysis]]
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| [[Category:Duality theories]]
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| [[Category:Integral representations]]
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