Riesz representation theorem

For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.

In functional analysis the Riesz representation theorem describes the dual of a Hilbert space. It is named in honour of Frigyes Riesz.

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 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, defined by

$\varphi_x (y) = \left\langle y , x \right\rangle \quad \forall y \in H$

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 Φ: H → H* defined by Φ(x) = φ_x is an isometric (anti-) isomorphism, meaning that:

• Φ is bijective.
• The norms of x and Φ(x) agree: $\Vert x \Vert = \Vert\Phi(x)\Vert$.
• Φ 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 Φ can be described as follows. Given an element φ of H*, the orthogonal complement of the kernel of φ 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 Φ(x) = φ.

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.

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.
• J. D. Gray, The shaping of the Riesz representation theorem: A chapter in the history of analysis, Archive for History in the Exact Sciences, Vol 31(2) 1984–85, 127–187.
• 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).
• D. G. Hartig, The Riesz representation theorem revisited, American Mathematical Monthly, 90(4), 277–280 (A category theoretic presentation as natural transformation).
• Hazewinkel, Michiel, ed. (2001), "Riesz representation theorem", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
• Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.
• Weisstein, Eric W., "Riesz Representation Theorem", MathWorld.
• Proof of Riesz representation theorem for separable Hilbert spaces, PlanetMath.org.
• Proof of Riesz representation theorem in Hilbert spaces on Bourbawiki