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

$\varphi_x(y) = \left\langle y , x \right\rangle,$

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.

Theorem. The mapping $\Phi$: H → H* defined by $\Phi(x)$ = $\varphi_x$ is an isometric (anti-) isomorphism, meaning that:

• $\Phi$ is bijective.
• The norms of $x$ and $\varphi_x$ agree: $\Vert x \Vert = \Vert\Phi(x)\Vert$.
• $\Phi$ is additive: $\Phi( x_1 + x_2 ) = \Phi( x_1 ) + \Phi( x_2 )$.
• If the base field is R, then $\Phi(\lambda x) = \lambda \Phi(x)$ for all real numbers λ.
• If the base field is C, then $\Phi(\lambda x) = \bar{\lambda} \Phi(x)$ for all complex numbers λ, where $\bar{\lambda}$ denotes the complex conjugation of λ.

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

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 $|\psi\rangle$ has a corresponding bra $\langle\psi|$, 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).
• Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.
• Proof of Riesz representation theorem for separable Hilbert spaces at PlanetMath.org.