# 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 φx, for all y in H defined by

$\varphi _{x}(y)=\left\langle y,x\right\rangle ,$ where $\langle \cdot ,\cdot \rangle$ 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.

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 $|\psi \rangle$ has a corresponding bra $\langle \psi |$ , and the correspondence is unambiguous. cf. also Rigged Hilbert space