Riesz representation theorem

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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[edit]

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

\phi_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 Φ: HH* defined by Φ(x) = φx 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 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 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. cf. also Rigged Hilbert space

References[edit]

  • 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).