Riesz representation theorem
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, 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 Φ: H → H* defined by Φ(x) = φx is an isometric (anti-) isomorphism, meaning that:
- Φ is bijective.
- The norms of x and Φ(x) agree: .
- Φ 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 Φ can be described as follows. Given a non-zero 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 . Then Φ(x) = φ.
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.
- 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.