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.

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 $\mathbb{R}$ or $\mathbb{C}$. If $x$ is an element of $H$, then the function $\varphi_x$, 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

$\Phi:H \rightarrow H^*, \quad \Phi(x) = \varphi_x$

is an isometric (anti-) isomorphism, meaning that:

• $\Phi$ is bijective.
• The norms of $x$ and $\Phi(x)$ 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 $\mathbb{R}$, then $\Phi(\lambda x) = \lambda \Phi(x)$ for all real numbers $\lambda$.
• If the base field is $\mathbb{C}$, then $\Phi(\lambda x) = \bar{\lambda} \Phi(x)$ for all complex numbers $\lambda$, where $\bar{\lambda}$ denotes the complex conjugation of $\lambda$.

The inverse map of $\Phi$ can be described as follows. Given an 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.

The representation theorem for linear functionals on $C_c (X)$

The following theorem represents positive linear functionals on $C_c (X)$, the space of continuous compactly supported complex-valued functions on a locally compact Hausdorff space $X$. The Borel sets in the following statement refer to the σ-algebra generated by the open sets.

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

• $\mu (K) < \infty$ 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 $\mu (E) < \infty$.

Theorem. Let X be a locally compact Hausdorff space. For any positive linear functional ψ on C_c(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 C_c(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.

Historical remark: In its original form by F. Riesz (1909) the theorem states that every continuous linear functional $A[f]$ over the space C[0,1] of continuous functions in the interval [0,1] can be represented in the form

$A[f] = \int_{0}^{1} f(x)\,d\alpha(x).$

where $\alpha(x)$ is a function of bounded variation on the interval [0,1], and the integral is a Riemann-Stieltjes integral. Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation (that assigns to each function of bounded variation the corresponding Lebesgue-Stieltjes measure, and the integral with respect to the Lebesgue-Stieltjes measure agrees with the Riemann-Stieltjes integral for continuous functions ), the above stated theorem generalizes the original statement of F. Riesz.

(See Gray(1984), for a historical discussion).

The representation theorem for the dual of $C_0 (X)$

The following theorem, also referred to as the Riesz-Markov theorem, gives a concrete realisation of the dual space of $C_0 (X)$, the set of continuous functions on $X$ which vanish at infinity. The Borel sets in the statement of the theorem also refers to the $\sigma$-algebra generated by the open sets.

If $\mu$ is a complex-valued countably additive Borel measure, $\mu$ is regular iff the non-negative countably additive measure $|\mu|$ is regular as defined above.

Theorem. Let $X$ be a locally compact Hausdorff space. For any continuous linear functional $\psi$ on $C_0 (X)$, there is a unique regular countably additive complex Borel measure $\mu$ on $X$ such that

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

for all $f$ in $C_0 (X)$. The norm of $\psi$ as a linear functional is the total variation of $\mu$, that is

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

Finally, $\psi$ is positive iff the measure $\mu$ is non-negative.

Remark. One might expect that by the Hahn-Banach theorem for bounded linear functionals, every bounded linear functional on $C_c (X)$ extends in exactly one way to a bounded linear functional on $C_0 (X)$, the latter being the closure of $C_c (X)$ in the supremum norm, and that for this reason the first statement implies the second. However the first result is for positive linear functionals, not bounded linear functionals, so the two facts are not equivalent.

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).
• Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.

• Weisstein, Eric W., "Riesz Representation Theorem" from MathWorld.
• Proof of Riesz representation theorem for separable Hilbert spaces on PlanetMath
• Proof of Riesz representation theorem in Hilbert spaces on Bourbawiki