There are several well-known theorems in functional analysis known as the Riesz representation theorem. Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...

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 (anti-) isomorphism is a particular natural one as will be described next. In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ...

Let H be a Hilbert space, and let H ' denote its dual space, consisting of all continuous linear operators from H into the base field R or C. If x is an element of H, then the function φ_x defined by In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. ...

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: // Definition Inner Product of two vectors Given twoN-by-1 column vectors v and u, the inner product is defined as the scalar quantity α resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ...

Theorem. The mapping

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. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...

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(x)], one can construct the bounded variational function α(x), such that, whatever the continuous function f(x) is, one has " Frigyes Riesz Frigyes Riesz (January 22, 1880 – February 28, 1956) was a mathematician who was born in GyÅ‘r, Austria-Hungary (now Hungary) and died in Budapest Hungary. ... Maurice Fréchet (born September 2, 1878, died June 4, 1973) was a French mathematician. ... 1907 (MCMVII) was a common year starting on Tuesday (see link for calendar) of the Gregorian calendar (or a common year starting on Wednesday of the 13-day-slower Julian calendar). ...

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. 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. For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard... In mathematics, a nuclear space is a topological vector space with many of the good properties of finite dimensional vector spaces. ...

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 complex valued functions of compact support. The Borel sets in the following statement refers to the σ-algebra generated by the open sets. In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space is either of two σ-algebras on a topological space X: The minimal σ-algebra containing the open sets. ...

A non-negative countably additive Borel measure μ on a locally compact Hausdorff space X is regular iff In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

• μ(K) < ∞ for every compact K;

• For every Borel set E,

• The relation

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 C_c(X), there is a unique Borel regular measure μ on X such that In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In functional analysis, a linear functional f on a C-star algebra is positive if whenever A is a positive element of . ... In mathematics, an outer measure on is Borel regular if for each there exists a Borel set such that and . ...

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. In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a Radon measure on a Hausdorff topological space X is a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular. ... Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

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. In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, a function on a normed vector space is said to vanish at infinity if as For example, the function defined on the real line vanishes at infinity. ... In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space is either of two σ-algebras on a topological space X: The minimal σ-algebra containing the open sets. ...

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 In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ...

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

Finally, ψ is positive iff the measure μ is non-negative. In functional analysis, a linear functional f on a C-star algebra is positive if whenever A is a positive element of . ...

Remark. A positive linear functional on C_c(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.
• 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, Amer. Math. Monthly, 90(4), 277-280 (category theoretic presentation as natural transformation)
• Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.

See also

Frigyes Riesz Frigyes Riesz Frigyes Riesz (January 22, 1880 – February 28, 1956) was a mathematician who was born in Győr, Austria-Hungary (now Hungary) and died in Budapest Hungary. ...

External links

• Mathworld
• Proof of Riesz representation theorem for separable Hilbert spaces on PlanetMath