# Riesz representation theorem

There are several well-known theorems in functional analysis known as the **Riesz representation theorem**. They are named in honour 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.

Let *H* be a Hilbert space, and let *H** denote its dual space, consisting of all continuous linear functionals from *H* into the field **R** or **C**. If *x* is an element of *H*, then the function φ_{x}, for all *y* 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.

**Theorem**. The mapping : *H* → *H** defined by (*x*) = _{x} is an isometric (anti-) isomorphism, meaning that:

- is bijective.
- The norms of
*x*and (*x*) agree: . - is additive: .
- If the base field is
**R**, then for all real numbers λ. - If the base field is
**C**, then for all complex numbers λ, where denotes the complex conjugation of λ.

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 (*x*) = .

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. When the theorem holds, every ket has a corresponding bra , and the correspondence is unambiguous. cf. also Rigged Hilbert space

## 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. - Template:Planetmath reference