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.

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[edit] The Hilbert space representation theorem

This theorem establishes an important connection between a Hilbert space and its 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 operators from H into the field \mathbb{R} or \mathbb{C}. If x is an element of H, then the function φx defined by

\phi_x(y) = \left\langle x , y \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

\Phi:H \rightarrow H^*, \quad \Phi(x) = \phi_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: Φ(x1 + x2) = Φ(x1) + Φ(x2).
  • If the base field is \mathbb{R}, then Φ(λx) = λΦ(x) for all real numbers λ.
  • If the base field is \mathbb{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 = \phi(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). Gray (1984) starts his review on the development up to the Riesz representation theorem with what he considers the pristine form in Riesz (1909): "Given the operation A[f], one can construct the function of bounded variation α(x), such that, whatever the continuous function f(x) is, one has A[f] = \int_{0}^{1} f(x)\,d\alpha(x)."

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. However, there are topological vector spaces, such as nuclear spaces, where the Riesz repesentation theorem does not hold, in which case the bra-ket notation can become awkward.

[edit] The representation theorem for linear functionals on Cc(X)

The following theorem represents positive linear functionals on Cc(X), the space of continuous complex valued functions of compact support. The Borel sets in the following statement refers to the σ-algebra generated by the open sets.

A non-negative countably additive Borel measure μ on a locally compact Hausdorff space X is regular if and only if

  • μ(K) < ∞ for every compact K;
  • For every Borel set E,
\mu(E) = \inf \{\mu(U): E \subseteq U, U \mbox{ open}\}
  • The relation
\mu(E) = \sup \{\mu(K): K \subseteq E, K \mbox{ compact}\}

holds whenever E is open or when E is Borel and μ(E) < ∞.

Theorem. Let X be a locally compact Hausdorff space. For any positive linear functional ψ on Cc(X), there is a unique Borel regular measure μ on X such that

\psi(f) = \int_X f(x) \, d \mu(x) \quad

for all f in Cc(X).

One approach to measure theory is to start with a Radon measure, defined as a positive linear functional on C(X). This is the way adopted by Bourbaki; it does of course assume that X starts life as a topological space, rather than simply as a set. For locally compact spaces an integration theory is then recovered.

[edit] The representation theorem for the dual of C0(X)

The following theorem, also referred to as the Riesz-Markov theorem gives a concrete realisation of the dual space of C0(X), the set of continuous functions on X which vanish at infinity. The Borel sets in the statement of the theorem also refers to the σ-algebra generated by the open sets. This result is similar to the result of the preceding section, but it does not subsume the previous result. See the technical remark below.

If μ is a complex-valued countably additive Borel measure, μ is regular if and only if the non-negative countably additive measure |μ| is regular as defined above.

Theorem. Let X be a locally compact Hausdorff space. For any continuous linear functional ψ on C0(X), there is a unique regular countably additive complex Borel measure μ on X such that

\psi(f) = \int_X f(x) \, d \mu(x) \quad

for all f in C0(X). The norm of ψ as a linear functional is the total variation of μ, that is

\|\psi\| = |\mu|(X).

Finally, ψ is positive if and only if the measure μ is non-negative.

Remark. A positive linear functional on Cc(X) may not extend to a bounded linear functional on C0(X). For this reason the previous results apply to slightly different situations.

[edit] 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 analytiques des systèmes de fonctions sommables. C. R. Acad. Sci. Paris 144, 1409–1411.
  • F. Riesz (1909). Sur les opérations fonctionelles 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(3) 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).
  • Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.
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