# Riesz representation theorem

There are several well-known theorems in functional analysis known as the Riesz representation theorem.

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## The Hilbert space representation theorem

This theorem establishes an important connection between a Hilbert space and its dual space: if the ground field is the real numbers, the two are isometrically isomorphic; if the ground field is the complex numbers, the two are isometrically anti-isomorphic. The theorem is the justification for the bra-ket notation popular in the mathematical treatment of quantum mechanics. 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 functions from H into the base field R or C. If x is an element of H, then the function φx defined by

[itex] \phi_x(y) = \langle x | y\rangle \quad \forall y \in H [itex]

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

[itex] \Phi:H \rightarrow H', \quad \Phi(x) = \phi_x [itex]

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

• Φ is bijective.
• The norms of x and Φ(x) agree: ||x|| = ||Φ(x)||.
• Φ is additive: Φ(x1 + x2) = Φ(x1) + Φ(x2).
• If the base field is R, then Φ(λ x) = λ Φ(x) for all real numbers λ.
• If the base field is C, then Φ(λ x) = λ* Φ(x) for all complex numbers λ, where λ* 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 = φ(z) / ||z||2 · z. Then Φ(x) = φ.

The theorem was proven simultaneously by Riesz and Fréchet in 1907 (see references).

## 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 iff

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

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 regular countably additive Borel measure μ on X such that

[itex] \psi(f) = \int_X f(x) d \mu(x) \quad [itex]

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.

## 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 iff 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

[itex] \psi(f) = \int_X f(x) d \mu(x) \quad [itex]

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

[itex] \|\psi\| = |\mu|(X).[itex]

Finally, ψ is positive iff 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.

## References

• M. Fréchet (1907). Sur les ensembles de fonctions et les opérations linéaires. C. R. Acad. Sci. Paris 144, 1414–1416.
• P. Halmos Measure Theory, D. van Nostrand and Co., 1950.
• F. Riesz (1907). Sur une espèce de géométrie analytiques des systèms de fonctions summable. C. R. Acad. Sci. Paris 144, 1409–1411.
• Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.

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