# Riesz representation theorem

The Riesz representation theorem is the name shared by several related results in functional analysis, the central one stating that every continuous linear functional on a [Hilbert space](https://www.edgepedia.org/hilbert-space) can be written as the inner product with a unique vector of that space. In this form it is also called the Fréchet–Riesz theorem, and it identifies a Hilbert space with its dual, the space of continuous linear functionals on it<sup>[1](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/8fb8d5c170f1613151aca71de21027bc_MIT18_102s21_full_lec.pdf)</sup><sup> • </sup><sup>[2](https://link.springer.com/article/10.1007/s00283-026-10563-w)</sup>. The same name covers a family of representation theorems for functionals on spaces of functions, culminating in the Riesz–Markov–Kakutani theorem, which represents positive linear functionals on spaces of continuous functions by measures<sup>[2](https://link.springer.com/article/10.1007/s00283-026-10563-w)</sup><sup> • </sup><sup>[3](https://encyclopediaofmath.org/wiki/Riesz_representation_theorem)</sup>.

| Key fact | Detail |
|---|---|
| Hilbert-space statement | For a Hilbert space \( H \), every \( f \in H' \) equals \( u \mapsto \langle u, v \rangle \) for a unique \( v \in H \)<sup>[1](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/8fb8d5c170f1613151aca71de21027bc_MIT18_102s21_full_lec.pdf)</sup> |
| First proof | Frigyes Riesz, 1907; the result is now often called the Fréchet–Riesz theorem<sup>[2](https://link.springer.com/article/10.1007/s00283-026-10563-w)</sup> |
| Riesz map | The map \( R : H^* \to H \), \( \lambda \mapsto g_\lambda \), is an antilinear isometric bijection, with \( \|\lambda\|_{H^*} = \|g_\lambda\|_H \)<sup>[4](https://num.math.uni-goettingen.de/schaback/teaching/AV_2.pdf)</sup> |
| Riesz–Stieltjes version (1909) | Continuous linear functionals on continuous functions over a compact interval are Riemann–Stieltjes integrals against functions of bounded variation<sup>[2](https://link.springer.com/article/10.1007/s00283-026-10563-w)</sup> |
| Measure version (1937) | Positive continuous linear functionals on \( \mathcal{C}(K) \) for compact \( K \) correspond to regular Borel measures; the general form is the Riesz–Markov–Kakutani theorem<sup>[2](https://link.springer.com/article/10.1007/s00283-026-10563-w)</sup> |
| \( L^p \) duality (1910–1916) | For \( 1 < p, q < \infty \) with \( 1/p + 1/q = 1 \), every continuous linear functional on \( L^p \) is integration against a function in \( L^q \)<sup>[2](https://link.springer.com/article/10.1007/s00283-026-10563-w)</sup> |

## The Hilbert-space theorem

Let \( H \) be a Hilbert space, a complete vector space with an inner product \( \langle \cdot, \cdot \rangle \). The theorem states that the two following conditions on a map \( \Phi \) are equivalent: \( \Phi \) is a continuous linear function from \( H \) to the real or complex numbers, and there is a unique member \( v \) of \( H \) such that \( \Phi(x) = \langle x, v \rangle \) for each \( x \) in \( H \)<sup>[5](https://harrell.math.gatech.edu/pde/Rieszapp.html)</sup>. In words, every continuous linear functional comes from an inner product with a fixed vector<sup>[5](https://harrell.math.gatech.edu/pde/Rieszapp.html)</sup>.

**How the proof works.** The standard argument rests on the projection lemma: if \( M \) is a closed subspace of a Hilbert space \( H \), every vector \( x \) decomposes uniquely as \( x = y + z \) with \( y \in M \) and \( z \) orthogonal to all of \( M \)<sup>[5](https://harrell.math.gatech.edu/pde/Rieszapp.html)</sup>. For a nonzero functional \( f \), its nullspace \( N = \{ w : f(w) = 0 \} \) is a closed subspace, and one shows \( N^{\perp} \) is one-dimensional, spanned by a single nonzero vector. Scaling a vector \( v_0 \) in that orthogonal complement produces the representing vector, and the inner product computation gives \( f(u) = \langle u, v \rangle \) for all \( u \)<sup>[1](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/8fb8d5c170f1613151aca71de21027bc_MIT18_102s21_full_lec.pdf)</sup>. Uniqueness follows because two candidates differ by a vector orthogonal to the whole space, hence zero<sup>[4](https://num.math.uni-goettingen.de/schaback/teaching/AV_2.pdf)</sup>.

Completeness of the space matters: the argument uses that the nullspace is closed and that orthogonal projections exist, properties tied to the Hilbert space structure rather than to an arbitrary inner-product space<sup>[5](https://harrell.math.gatech.edu/pde/Rieszapp.html)</sup>.

**The Riesz map.** Sending each functional \( \lambda \) to its representer \( g_\lambda \) defines the Riesz map \( R : H^* \to H \). It is an antilinear isometric bijection between \( H \) and its dual: antilinear because the inner product is conjugate-linear in one argument in the complex case, and isometric because \( \|\lambda\|_{H^*} = \|g_\lambda\|_H \), with equality in the Cauchy–Schwarz bound attained at \( u = g_\lambda \)<sup>[4](https://num.math.uni-goettingen.de/schaback/teaching/AV_2.pdf)</sup>. Thus every Hilbert space is isometrically isomorphic to its dual, a identification available in Banach spaces only in special cases such as \( L^p \) with \( p = 2 \)<sup>[4](https://num.math.uni-goettingen.de/schaback/teaching/AV_2.pdf)</sup>.

## The family of representation theorems

The name attaches to a sequence of results, each replacing the representing object with something suited to the function space involved<sup>[2](https://link.springer.com/article/10.1007/s00283-026-10563-w)</sup>.

**Stieltjes integrals, 1909.** In 1909 Riesz showed that continuous linear functionals on spaces of continuous functions over a compact interval are represented by Riemann–Stieltjes integrals \( A(f) = \int_a^b f(s) \, d\alpha(s) \) against functions \( \alpha \) of bounded variation. This replaced the coordinate vector of the Hilbert-space case with a function and the finite sum with an integral, and it is described as the first bridge between classical integral calculus and the emerging theory of measure<sup>[2](https://link.springer.com/article/10.1007/s00283-026-10563-w)</sup>. In the 1909 paper Riesz did not address the uniqueness of the function \( \alpha \) or its relation to the norm of the functional; once a normalization rule is fixed, the norm \( \|A\| \) equals \( |\alpha(a)| \) plus the total variation of \( \alpha \) on \( [a,b] \)<sup>[6](https://www.treccani.it/enciclopedia/la-seconda-rivoluzione-scientifica-matematica-e-logica-le-origini-dell-analisi-funzionale_(Storia-della-Scienza)/)</sup>.

**Lebesgue space duality, 1910–1916.** In his paper Über lineare Funktionalgleichungen, Riesz established that for \( 1 < p, q < \infty \) with \( 1/p + 1/q = 1 \), every continuous linear functional on \( L^p \) is given by integration against a function in \( L^q \), with vectors replaced by integrable functions and inner products by integral pairings<sup>[2](https://link.springer.com/article/10.1007/s00283-026-10563-w)</sup>.

**Measures, 1937.** In Sur les fonctions continues, Riesz proved that every positive and continuous linear functional on \( \mathcal{C}(K) \), the space of continuous real-valued functions on a compact space \( K \), corresponds uniquely to a regular [Borel measure](https://www.edgepedia.org/borel-measure) \( \mu \), with \( L(f) = \int_K f \, d\mu \)<sup>[2](https://link.springer.com/article/10.1007/s00283-026-10563-w)</sup><sup> • </sup><sup>[3](https://encyclopediaofmath.org/wiki/Riesz_representation_theorem)</sup>. The Encyclopedia of Mathematics describes this statement, sometimes called the Riesz–Markov theorem, as a central theorem of classical measure theory, often stated more generally for locally compact Hausdorff spaces<sup>[3](https://encyclopediaofmath.org/wiki/Riesz_representation_theorem)</sup>. A related statement identifies the dual of \( C(X) \) with the space \( \mathcal{M}^b(X) \) of real-valued measures of finite total variation; combined with the Radon–Nikodým theorem, it says every \( L \in (C(X))' \) can be written \( L(f) = \int_X f g \, d\mu \) with a [Radon measure](https://www.edgepedia.org/radon-measure) \( \mu \) and a Borel function \( g \) satisfying \( |g| = 1 \) almost everywhere<sup>[3](https://encyclopediaofmath.org/wiki/Riesz_representation_theorem)</sup>. More general versions for non-compact locally compact Hausdorff spaces, using continuous functions of compact support, are attributed to Kakutani<sup>[7](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/representing_measures.html)</sup>.

## Applications

**Adjoint operators.** The theorem gives the existence and uniqueness of the adjoint \( A^* \) of a bounded linear operator \( A : H \to H \), the operator satisfying \( \langle Au, v \rangle = \langle u, A^* v \rangle \) for all \( u, v \), and it yields the norm equality \( \|A^*\| = \|A\| \)<sup>[1](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/8fb8d5c170f1613151aca71de21027bc_MIT18_102s21_full_lec.pdf)</sup>. Because duals of Hilbert spaces are identified with the spaces themselves, adjoints behave like ordinary operators, the analogs of transposed matrices in finite dimensions<sup>[1](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/8fb8d5c170f1613151aca71de21027bc_MIT18_102s21_full_lec.pdf)</sup>.

**Reproducing kernel Hilbert spaces.** If \( H \) is a Hilbert space of functions on a set \( \Omega \) in which every point evaluation \( \delta_x : f \mapsto f(x) \) is continuous, the Riesz map sends \( \delta_x \) to a function \( K(x, \cdot) \), and the resulting kernel satisfies the reproduction equation \( f(x) = \langle f, K(x, \cdot) \rangle_H \). The Riesz representer of a functional \( \lambda \) in the dual is \( \lambda_x K(\cdot, x) \)<sup>[4](https://num.math.uni-goettingen.de/schaback/teaching/AV_2.pdf)</sup>.

**Partial differential equations.** The Lax–Milgram theorem, used heavily in the weak formulation of partial differential equations, generalizes Riesz representation by replacing the inner product with a bounded, coercive bilinear form \( a(\cdot, \cdot) \) paired with a bounded linear form \( l \)<sup>[8](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/functional_analysis.html)</sup>.

**Probability and measure.** A historical survey describes the sequence from Fréchet–Riesz through Riesz–Stieltjes and \( L^p \) duality to Riesz–Markov–Kakutani as the analytical framework underlying modern probability, in which expectation, distribution, conditional expectation, and the Wiener measure appear as manifestations of a single representation principle<sup>[2](https://link.springer.com/article/10.1007/s00283-026-10563-w)</sup>.

## History

[Frigyes Riesz](https://www.edgepedia.org/frigyes-riesz) (1880–1956), a Hungarian mathematician and a leading figure in the founding of functional analysis, proved the Hilbert-space representation result in 1907<sup>[2](https://link.springer.com/article/10.1007/s00283-026-10563-w)</sup><sup> • </sup><sup>[9](https://mathshistory.st-andrews.ac.uk/Biographies/Riesz/)</sup>. Earlier authors, notably Hadamard and Fréchet, had given representations for continuous linear functionals on \( C[0,1] \), but their results lacked the clarity, elegance, and uniqueness of Riesz's 1909 integral representation, after which the result became known as the Riesz representation theorem<sup>[10](https://digital.library.unt.edu/ark%3A/67531/metadc504232/)</sup>. The 1909 announcement appeared in the paper Sur les opérations fonctionnelles linéaires<sup>[11](https://nonagon.org/ExLibris/node/49)</sup>. Later work extended the principle to positive operators from function spaces into ordered vector spaces, generalizing the theorems for \( C_c(X) \) and \( C_0(X) \)<sup>[12](https://arxiv.org/abs/2104.12153)</sup>.

## References

1. [MIT 18.102 Introduction to Functional Analysis, Spring 2021, complete lecture notes](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/8fb8d5c170f1613151aca71de21027bc_MIT18_102s21_full_lec.pdf)
2. [From Riesz to Kakutani: Revealing Foundations of Probability via Representation Theorems, The Mathematical Intelligencer](https://link.springer.com/article/10.1007/s00283-026-10563-w)
3. [Riesz representation theorem, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Riesz_representation_theorem)
4. [R. Schaback, Numerische Mathematik lecture notes, Theorem 11.15 and Section 11.6, University of Göttingen](https://num.math.uni-goettingen.de/schaback/teaching/AV_2.pdf)
5. [E. M. Harrell II and J. V. Herod, Linear Methods of Applied Mathematics, The Riesz representation theorem](https://harrell.math.gatech.edu/pde/Rieszapp.html)
6. [La seconda rivoluzione scientifica: matematica e logica, le origini dell'analisi funzionale, Treccani](https://www.treccani.it/enciclopedia/la-seconda-rivoluzione-scientifica-matematica-e-logica-le-origini-dell-analisi-funzionale_(Storia-della-Scienza)/)
7. [Representing Measures, lecture notes, University of Manchester](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/representing_measures.html)
8. [Relevant concepts from functional analysis, MA8502, NTNU](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/functional_analysis.html)
9. [Frigyes Riesz (1880–1956), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Riesz/)
10. [The Riesz Representation Theorem, UNT Digital Library](https://digital.library.unt.edu/ark%3A/67531/metadc504232/)
11. [Riesz's Sur les opérations fonctionnelles linéaires, Ex Libris](https://nonagon.org/ExLibris/node/49)
12. [Riesz representation theorems for positive linear operators, arXiv:2104.12153](https://arxiv.org/abs/2104.12153)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Functional analysis*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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