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

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

where ${\displaystyle \langle \cdot ,\cdot \rangle }$ 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.

Theorem. The mapping ${\displaystyle \Phi }$: HH* defined by ${\displaystyle \Phi }$(x) = ${\displaystyle \varphi }$x is an isometric (anti-) isomorphism, meaning that:

The inverse map of ${\displaystyle \Phi }$ can be described as follows. Given a non-zero element ${\displaystyle \varphi }$ of H*, the orthogonal complement of the kernel of ${\displaystyle \varphi }$ is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set ${\displaystyle x={\overline {\varphi (z)}}\cdot z/{\left\Vert z\right\Vert }^{2}}$. Then ${\displaystyle \Phi }$(x) = ${\displaystyle \varphi }$.

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 ${\displaystyle |\psi \rangle }$ has a corresponding bra ${\displaystyle \langle \psi |}$, 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).