Riesz representation theorem

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There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honor 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[edit]

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; a natural isomorphism.

Let H be a Hilbert space, and let H* denote its dual space, consisting of all continuous linear functionals from H into the field or . If is an element of H, then the function for all in H defined by:

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.

Riesz–Fréchet representation theorem. Let be a Hilbert space and . Then there exists such that for any , . Moreover

Proof. Let . Clearly is closed subspace of . If , then we can trivially choose . Now assume . Then is one-dimensional. Indeed, let be nonzero vectors in . Then there is nonzero real number , such that . Observe that and , so . This means that . Now let be unit vector in . For arbitrary , let be the orthogonal projections of onto respectively. Then and (from the properties of orthogonal projections), so that and . Thus . Hence . We also see . From the Cauchy-Bunyakovsky-Schwartz inequality , thus for with unit norm . This implies that .

Given any continuous linear functional g in H*, the corresponding element can be constructed uniquely by , where is an orthonormal basis of H, and the value of does not vary by choice of basis. Thus, if , then

Theorem. The mapping : HH* defined by = is an isometric (anti-) isomorphism, meaning that:

  • is bijective.
  • The norms of and agree: .
  • is additive: .
  • If the base field is , then for all real numbers λ.
  • If the base field is , then for all complex numbers λ, where denotes the complex conjugation of .

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

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. The theorem says that, every bra has a corresponding ket , and the latter is unique.


  • 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.
  • 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).
  • Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.
  • "Proof of Riesz representation theorem for separable Hilbert spaces". PlanetMath.