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Riesz representation theorem

The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, 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 isomorphism.

This article is about a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem. For other theorems, see Riesz theorem.

 – Area of functional analysis and convex analysis

Choquet theory

 – Operator in probability theory

Covariance operator

Fundamental theorem of Hilbert spaces

 – Functions on special groups related to their matrix representations

Matrix coefficient

Bachman, George; Narici, Lawrence (2000). Functional Analysis (Second ed.). Mineola, New York: Dover Publications.  978-0486402512. OCLC 829157984.

ISBN

Fréchet, M. (1907). . Les Comptes rendus de l'Académie des sciences (in French). 144: 1414–1416.

"Sur les ensembles de fonctions et les opérations linéaires"

Measure Theory, D. van Nostrand and Co., 1950.

P. Halmos

P. Halmos, A Hilbert Space Problem Book, Springer, New York 1982 (problem 3 contains version for vector spaces with coordinate systems).

Riesz, F. (1907). . Comptes rendus de l'Académie des Sciences (in French). 144: 1409–1411.

"Sur une espèce de géométrie analytique des systèmes de fonctions sommables"

Riesz, F. (1909). . Comptes rendus de l'Académie des Sciences (in French). 149: 974–977.

"Sur les opérations fonctionnelles linéaires"