Riesz representation theorem
The Riesz representation theorem identifies continuous linear functionals on a Hilbert space with vectors in that space. The designation also applies to a related representation theorem that identifies positive linear functionals on spaces of continuous functions with measures. Although the two forms concern different categories of objects, both convert an abstract linear functional into integration against a concrete representing object.
In the Hilbert-space form, the representing object is a vector and the functional is expressed through the inner product. In the measure-theoretic form, the representing object is a regular measure and the functional is expressed through an integral. These results form a central connection among functional analysis, measure theory, and the theory of operator algebras.
Hilbert-space theorem
Let (H) be a real or complex Hilbert space with inner product (\langle \cdot,\cdot\rangle). Under the convention that the inner product is linear in its first argument, every continuous linear functional
[ \varphi:H\longrightarrow \mathbb F, ]
where (\mathbb F) is either (\mathbb R) or (\mathbb C), has a unique representation
[ \varphi(x)=\langle x,y\rangle ]
for some (y\in H). The representing vector satisfies
[ |\varphi|=|y|. ]
Consequently, the map from (H) to its continuous dual space defined by
[ y\longmapsto \bigl(x\longmapsto \langle x,y\rangle\bigr) ]
is an isometric isomorphism in the real case. In the complex case it is conjugate-linear under the stated convention, because scalar multiplication in the second argument of the inner product is conjugate-linear.
If the alternative convention is adopted, with the inner product linear in its second argument, the formula becomes (\varphi(x)=\langle y,x\rangle). The mathematical content remains unchanged, although the placement of the representing vector determines whether the identification with the dual is linear or conjugate-linear. A 1928 formulation by You Watanabe separated these two conventions by expressing the result as a canonical linear isometry from the conjugate Hilbert space (\overline H) onto (H^*). This formulation became part of the convention-independent treatment of complex Hilbert spaces.
Structure of the representation
For a nonzero functional (\varphi), its kernel
[ M=\ker\varphi ]
is a closed subspace of codimension one. The orthogonal decomposition theorem gives
[ H=M\oplus M^\perp. ]
Since (M^\perp) is one-dimensional, a nonzero vector (u\in M^\perp) determines the value of (\varphi) on the entire orthogonal complement. Every (x\in H) decomposes uniquely as (x=m+\alpha u), where (m\in M), and therefore
[ \varphi(x)=\alpha\varphi(u). ]
An appropriate scalar multiple of (u) then yields the representing vector (y). The zero functional is represented uniquely by the zero vector.
The norm identity follows directly from the Cauchy–Schwarz inequality, which gives
[ |\varphi(x)|=|\langle x,y\rangle| \leq |x|,|y|. ]
Thus (|\varphi|\leq |y|). Equality is obtained by evaluating on (y/|y|) when (y\neq 0), so the representation preserves the norm exactly. Uniqueness follows because two representing vectors (y) and (z) would satisfy
[ \langle x,y-z\rangle=0 ]
for every (x\in H), and substitution of (x=y-z) forces (y=z).
Canonical duality and self-duality
The theorem makes every Hilbert space isometrically identifiable with its continuous dual, subject in the complex case to conjugation. This form of self-duality is stronger than the canonical embedding of a general Banach space into its bidual. A Banach space need not be isomorphic to its dual, whereas the inner product on a Hilbert space supplies a representing vector for every bounded functional.
For a complex Hilbert space, the conjugate space (\overline H) has the same additive group and norm as (H), while scalar multiplication is defined by
[ \lambda\cdot_{\overline H}x=\overline{\lambda}x. ]
The correspondence
[ \overline H\longrightarrow H^*,\qquad y\longmapsto \langle,\cdot,,y\rangle ]
is then linear and isometric. This description avoids treating the conjugate-linearity of the usual map (H\to H^*) as an exceptional feature.
The canonical map from (H) into its bidual (H^{**}) is linear even over the complex field. Since (H^*) is itself a Hilbert space through the Riesz correspondence, the theorem implies that every Hilbert space is reflexive. Reflexivity here follows from the inner-product structure rather than from finite dimensionality.
Operator formulation
Let (H) and (K) be Hilbert spaces, and let
[ T:H\longrightarrow K ]
be a bounded linear operator. For each (y\in K), the expression
[ x\longmapsto \langle Tx,y\rangle_K ]
defines a continuous linear functional on (H). The Riesz theorem therefore determines a unique vector (T^*y\in H) satisfying
[ \langle Tx,y\rangle_K
\langle x,T^*y\rangle_H. ]
The resulting operator (T^*:K\to H) is the adjoint operator of (T). Its existence for every bounded operator between Hilbert spaces depends directly on the representation theorem.
This construction underlies the definitions of self-adjoint operators, normal operators, and unitary operators. It also converts questions about linear functionals into questions about vectors, allowing orthogonality and projection to be applied to dual-space problems without introducing a separate coordinate representation.
Riesz–Markov–Kakutani representation
A second theorem bearing Riesz’s name concerns positive linear functionals on spaces of continuous functions. Let (X) be a locally compact Hausdorff space, and let (C_0(X)) denote the Banach space of continuous complex-valued functions that vanish at infinity, equipped with the supremum norm. For every positive linear functional
[ \Lambda:C_0(X)\longrightarrow\mathbb C, ]
there exists a unique regular positive Borel measure (\mu) on (X) such that
[ \Lambda(f)=\int_X f,d\mu ]
for every (f\in C_0(X)). Moreover,
[ |\Lambda|=\mu(X), ]
where the right-hand side may be interpreted through the total mass of the representing measure.
For an arbitrary bounded linear functional on (C_0(X)), the representing object is a finite regular complex Borel measure. Its norm is the total variation:
[ |\Lambda|=|\mu|(X). ]
The correspondence identifies (C_0(X)^*) isometrically with the space (M(X)) of finite regular complex measures on (X).
Frigyes Riesz established an early form of this result for continuous functions on a compact interval, where the representing object could be expressed through a function of bounded variation and a Riemann–Stieltjes integral. Shizuo Kakutani later placed the representation in the locally compact Hausdorff setting, in which regular Borel measures provide the natural dual objects for (C_0(X)).
Regularity and uniqueness
Regularity connects the measure to the topology of (X). For a positive regular Borel measure, the measure of an open set is determined by compact subsets contained within it, while the measure of a Borel set is determined by open sets containing it. These approximation properties ensure that the values of the functional on continuous functions determine the measure uniquely.
In the compact case, (C_0(X)=C(X)), and the constant function (1) belongs to the function space. A positive functional then satisfies
[ |\Lambda|=\Lambda(1)=\mu(X). ]
For a noncompact locally compact space, the constant function need not vanish at infinity. The norm is instead recovered through compactly supported continuous functions that approximate the role of the constant function on successively larger compact subsets.
The theorem extends from positive functionals to signed or complex functionals by decomposition. A bounded real functional corresponds to a finite signed regular measure through the Jordan decomposition theorem, while a bounded complex functional corresponds to a complex measure whose total variation controls the functional norm.
Relation between the two representation theorems
The Hilbert-space and measure-theoretic theorems share a common representational pattern but use different structures. In a Hilbert space, the inner product identifies a functional with a vector belonging to the original space. In (C_0(X)), the supremum norm does not generally arise from an inner product, so a functional is represented by a measure rather than by another continuous function.
When a Hilbert space is realized as an (L^2) space, the Hilbert-space theorem gives
[ \varphi(f)=\int_X f,\overline g,d\mu ]
for a unique (g\in L^2(X,\mu)). This representation concerns functionals continuous in the (L^2)-norm. By contrast, a functional on (C_0(X)) is continuous in the supremum norm and is represented by a measure that need not possess a density with respect to any previously selected measure.
If the representing measure is absolutely continuous with respect to a reference measure, the Radon–Nikodym theorem supplies a density. The functional may then be written as integration against that density, although the applicable integrability class depends on the normed function space under consideration.
Historical development
Frigyes Riesz developed the foundational representation results during the early formation of functional analysis. His work connected linear functionals with integral expressions and established the geometric role of orthogonality in spaces modeled on square-summable functions. The later abstraction of complete inner-product spaces produced the modern Hilbert-space statement.
The terminology of Hilbert space became standard through the abstract operator-theoretic framework associated with John von Neumann. Within that framework, the representation theorem supplied the duality required for adjoints and orthogonal projections. The measure form developed through subsequent generalizations from compact intervals to topological spaces, culminating in the formulation commonly called the Riesz–Markov–Kakutani representation theorem.