# Riesz representation theorem

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

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 projection of onto . 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 : *H* → *H** 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.

## 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. - 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*.