The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, 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 underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism.

Preliminaries and notation

Let H be a Hilbert space over a field \mathbb{F}, where \mathbb{F} is either the real numbers \R or the complex numbers \Complex. If (resp. if ) then H is called a (resp. a ). Every real Hilbert space can be extended to be a dense subset of a unique (up to bijective isometry) complex Hilbert space, called its complexification, which is why Hilbert spaces are often automatically assumed to be complex. Real and complex Hilbert spaces have in common many, but by no means all, properties and results/theorems. This article is intended for both mathematicians and physicists and will describe the theorem for both. In both mathematics and physics, if a Hilbert space is assumed to be real (that is, if ) then this will usually be made clear. Often in mathematics, and especially in physics, unless indicated otherwise, "Hilbert space" is usually automatically assumed to mean "complex Hilbert space." Depending on the author, in mathematics, "Hilbert space" usually means either (1) a complex Hilbert space, or (2) a real complex Hilbert space.

Linear and antilinear maps

By definition, an Antilinear map (also called a ) f : H \to Y is a map between vector spaces that is : and (also called or ): where is the conjugate of the complex number c = a + b i, given by. In contrast, a map f : H \to Y is linear if it is additive and Homogeneous function: Every constant 0 map is always both linear and antilinear. If then the definitions of linear maps and antilinear maps are completely identical. A linear map from a Hilbert space into a Banach space (or more generally, from any Banach space into any topological vector space) is continuous if and only if it is bounded; the same is true of antilinear maps. The inverse of any antilinear (resp. linear) bijection is again an antilinear (resp. linear) bijection. The composition of two linear maps is a map. Continuous dual and anti-dual spaces A on H is a function whose codomain is the underlying scalar field \mathbb{F}. Denote by H^* (resp. by the set of all continuous linear (resp. continuous antilinear) functionals on H, which is called the (resp. the ) of H. If then linear functionals on H are the same as antilinear functionals and consequently, the same is true for such continuous maps: that is, One-to-one correspondence between linear and antilinear functionals Given any functional the is the functional This assignment is most useful when because if then and the assignment reduces down to the identity map. The assignment defines an antilinear bijective correspondence from the set of onto the set of

Mathematics vs. physics notations and definitions of inner product

The Hilbert space H has an associated inner product valued in H's underlying scalar field \mathbb{F} that is linear in one coordinate and antilinear in the other (as specified below). If H is a complex Hilbert space, then there is a crucial difference between the notations prevailing in mathematics versus physics, regarding which of the two variables is linear. However, for real Hilbert spaces, the inner product is a symmetric map that is linear in each coordinate (bilinear), so there can be no such confusion. In mathematics, the inner product on a Hilbert space H is often denoted by or while in physics, the bra–ket notation or is typically used. In this article, these two notations will be related by the equality: These have the following properties:<ol>

<li>The map is linear in its first coordinate; equivalently, the map is linear in its second coordinate. That is, for fixed y \in H, the map with is a linear functional on H. This linear functional is continuous, so </li> <li>The map is [antilinear](https://bliptext.com/articles/antilinear-map) in its coordinate; equivalently, the map is [antilinear](https://bliptext.com/articles/antilinear-map) in its coordinate. That is, for fixed y \in H, the map with is an antilinear functional on H. This antilinear functional is continuous, so </li> </ol> In computations, one must consistently use either the mathematics notation, which is (linear, antilinear); or the physics notation , whch is (antilinear | linear).

Canonical norm and inner product on the dual space and anti-dual space

If x = y then is a non-negative real number and the map defines a canonical norm on H that makes H into a normed space. As with all normed spaces, the (continuous) dual space H^* carries a canonical norm, called the, that is defined by The canonical norm on the (continuous) anti-dual space denoted by is defined by using this same equation: This canonical norm on H^* satisfies the parallelogram law, which means that the polarization identity can be used to define a which this article will denote by the notations where this inner product turns H^* into a Hilbert space. There are now two ways of defining a norm on H^: the norm induced by this inner product (that is, the norm defined by ) and the usual dual norm (defined as the supremum over the closed unit ball). These norms are the same; explicitly, this means that the following holds for every f \in H^: As will be described later, the Riesz representation theorem can be used to give an equivalent definition of the canonical norm and the canonical inner product on H^. The same equations that were used above can also be used to define a norm and inner product on H's anti-dual space Canonical isometry between the dual and antidual The complex conjugate of a functional f, which was defined above, satisfies for every f \in H^ and every This says exactly that the canonical antilinear bijection defined by as well as its inverse are antilinear isometries and consequently also homeomorphisms. The inner products on the dual space H^* and the anti-dual space denoted respectively by and are related by and If then and this canonical map reduces down to the identity map.

Riesz representation theorem

Two vectors x and y are if which happens if and only if for all scalars s. The orthogonal complement of a subset is which is always a closed vector subspace of H. The Hilbert projection theorem guarantees that for any nonempty closed convex subset C of a Hilbert space there exists a unique vector m \in C such that that is, m \in C is the (unique) global minimum point of the function defined by

Statement

Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907 (see references). Let \mathbb{F} denote the underlying scalar field of H. Fix y \in H. Define by which is a linear functional on H since z is in the linear argument. By the Cauchy–Schwarz inequality, which shows that \Lambda is bounded (equivalently, continuous) and that It remains to show that By using y in place of z, it follows that (the equality holds because is real and non-negative). Thus that The proof above did not use the fact that H is complete, which shows that the formula for the norm holds more generally for all inner product spaces. Suppose f, g \in H are such that and for all z \in H. Then which shows that is the constant 0 linear functional. Consequently which implies that f - g = 0. Let If K = H (or equivalently, if \varphi = 0) then taking completes the proof so assume that K \neq H and The continuity of \varphi implies that K is a closed subspace of H (because and { 0 } is a closed subset of \mathbb{F}). Let denote the orthogonal complement of K in H. Because K is closed and H is a Hilbert space, H can be written as the direct sum (a proof of this is given in the article on the Hilbert projection theorem). Because K \neq H, there exists some non-zero For any h \in H, which shows that where now implies Solving for \varphi h shows that which proves that the vector satisfies Applying the norm formula that was proved above with shows that Also, the vector has norm |u| = 1 and satisfies It can now be deduced that K^{\bot} is 1-dimensional when Let be any non-zero vector. Replacing p with q in the proof above shows that the vector satisfies for every h \in H. The uniqueness of the (non-zero) vector f_{\varphi} representing \varphi implies that which in turn implies that and Thus every vector in K^{\bot} is a scalar multiple of The formulas for the inner products follow from the polarization identity.

Observations

If then So in particular, is always real and furthermore, if and only if if and only if Linear functionals as affine hyperplanes A non-trivial continuous linear functional \varphi is often interpreted geometrically by identifying it with the affine hyperplane (the kernel is also often visualized alongside although knowing A is enough to reconstruct because if then and otherwise ). In particular, the norm of \varphi should somehow be interpretable as the "norm of the hyperplane A". When then the Riesz representation theorem provides such an interpretation of |\varphi| in terms of the affine hyperplane as follows: using the notation from the theorem's statement, from it follows that and so implies and thus This can also be seen by applying the Hilbert projection theorem to A and concluding that the global minimum point of the map defined by is The formulas provide the promised interpretation of the linear functional's norm |\varphi| entirely in terms of its associated affine hyperplane (because with this formula, knowing only the A is enough to describe the norm of its associated linear ). Defining the infimum formula will also hold when When the supremum is taken in \R (as is typically assumed), then the supremum of the empty set is but if the supremum is taken in the non-negative reals [0, \infty) (which is the image/range of the norm when \dim H > 0) then this supremum is instead in which case the supremum formula will also hold when \varphi = 0 (although the atypical equality is usually unexpected and so risks causing confusion).

Constructions of the representing vector

Using the notation from the theorem above, several ways of constructing f_{\varphi} from are now described. If \varphi = 0 then ; in other words, f_0 = 0. This special case of \varphi = 0 is henceforth assumed to be known, which is why some of the constructions given below start by assuming Orthogonal complement of kernel If then for any If is a unit vector (meaning |u| = 1) then (this is true even if \varphi = 0 because in this case ). If u is a unit vector satisfying the above condition then the same is true of -u, which is also a unit vector in However, so both these vectors result in the same Orthogonal projection onto kernel If x \in H is such that and if x_K is the orthogonal projection of x onto \ker\varphi then Orthonormal basis Given an orthonormal basis of H and a continuous linear functional the vector can be constructed uniquely by where all but at most countably many will be equal to 0 and where the value of f_{\varphi} does not actually depend on choice of orthonormal basis (that is, using any other orthonormal basis for H will result in the same vector). If y \in H is written as then and If the orthonormal basis is a sequence then this becomes and if y \in H is written as then

Example in finite dimensions using matrix transformations

Consider the special case of (where n > 0 is an integer) with the standard inner product where are represented as column matrices and with respect to the standard orthonormal basis on H (here, e_i is 1 at its ith coordinate and 0 everywhere else; as usual, H^* will now be associated with the dual basis) and where denotes the conjugate transpose of \vec{z}. Let be any linear functional and let be the unique scalars such that where it can be shown that for all Then the Riesz representation of \varphi is the vector To see why, identify every vector in H with the column matrix so that f_{\varphi} is identified with As usual, also identify the linear functional \varphi with its transformation matrix, which is the row matrix so that and the function \varphi is the assignment where the right hand side is matrix multiplication. Then for all which shows that f_{\varphi} satisfies the defining condition of the Riesz representation of \varphi. The bijective antilinear isometry defined in the corollary to the Riesz representation theorem is the assignment that sends to the linear functional on H defined by where under the identification of vectors in H with column matrices and vector in H^* with row matrices, \Phi is just the assignment As described in the corollary, \Phi's inverse is the antilinear isometry which was just shown above to be: where in terms of matrices, \Phi^{-1} is the assignment Thus in terms of matrices, each of and is just the operation of conjugate transposition (although between different spaces of matrices: if H is identified with the space of all column (respectively, row) matrices then H^* is identified with the space of all row (respectively, column matrices). This example used the standard inner product, which is the map but if a different inner product is used, such as where M is any Hermitian positive-definite matrix, or if a different orthonormal basis is used then the transformation matrices, and thus also the above formulas, will be different.

Relationship with the associated real Hilbert space

Assume that H is a complex Hilbert space with inner product When the Hilbert space H is reinterpreted as a real Hilbert space then it will be denoted by H_{\R}, where the (real) inner-product on H_{\R} is the real part of H's inner product; that is: The norm on H_{\R} induced by is equal to the original norm on H and the continuous dual space of H_{\R} is the set of all -valued bounded \R-linear functionals on H_{\R} (see the article about the polarization identity for additional details about this relationship). Let and denote the real and imaginary parts of a linear functional \psi, so that The formula expressing a linear functional in terms of its real part is where for all h \in H. It follows that and that \psi = 0 if and only if It can also be shown that where and are the usual operator norms. In particular, a linear functional \psi is bounded if and only if its real part \psi_{\R} is bounded. Representing a functional and its real part The Riesz representation of a continuous linear function \varphi on a complex Hilbert space is equal to the Riesz representation of its real part on its associated real Hilbert space. Explicitly, let and as above, let be the Riesz representation of \varphi obtained in so it is the unique vector that satisfies for all x \in H. The real part of \varphi is a continuous real linear functional on H_{\R} and so the Riesz representation theorem may be applied to and the associated real Hilbert space to produce its Riesz representation, which will be denoted by That is, is the unique vector in H_{\R} that satisfies for all x \in H. The conclusion is This follows from the main theorem because and if x \in H then and consequently, if then which shows that Moreover, being a real number implies that In other words, in the theorem and constructions above, if H is replaced with its real Hilbert space counterpart H_{\R} and if \varphi is replaced with then This means that vector f_{\varphi} obtained by using and the real linear functional is the equal to the vector obtained by using the origin complex Hilbert space and original complex linear functional \varphi (with identical norm values as well). Furthermore, if then f_{\varphi} is perpendicular to with respect to where the kernel of \varphi is be a proper subspace of the kernel of its real part Assume now that Then because and \ker\varphi is a proper subset of The vector subspace has real codimension 1 in while has codimension 1 in H_{\R}, and That is, f_{\varphi} is perpendicular to with respect to

Canonical injections into the dual and anti-dual

Induced linear map into anti-dual The map defined by placing y into the coordinate of the inner product and letting the variable h \in H vary over the coordinate results in an functional: This map is an element of which is the continuous anti-dual space of H. The is the operator which is also an injective isometry. The Fundamental theorem of Hilbert spaces, which is related to Riesz representation theorem, states that this map is surjective (and thus bijective). Consequently, every antilinear functional on H can be written (uniquely) in this form. If is the canonical linear bijective isometry that was defined above, then the following equality holds:

Extending the bra–ket notation to bras and kets

Let be a Hilbert space and as before, let Let which is a bijective antilinear isometry that satisfies Bras Given a vector h \in H, let denote the continuous linear functional \Phi h; that is, so that this functional is defined by This map was denoted by earlier in this article. The assignment is just the isometric antilinear isomorphism which is why holds for all g, h \in H and all scalars c. The result of plugging some given g \in H into the functional is the scalar which may be denoted by Bra of a linear functional Given a continuous linear functional let denote the vector ; that is, The assignment is just the isometric antilinear isomorphism which is why holds for all and all scalars c. The defining condition of the vector is the technically correct but unsightly equality which is why the notation is used in place of With this notation, the defining condition becomes Kets For any given vector g \in H, the notation is used to denote g; that is, The assignment is just the identity map which is why holds for all g, h \in H and all scalars c. The notation and is used in place of and respectively. As expected, and really is just the scalar

Adjoints and transposes

Let A : H \to Z be a continuous linear operator between Hilbert spaces and As before, let and Denote by the usual bijective antilinear isometries that satisfy:

Definition of the adjoint

For every z \in Z, the scalar-valued map on H defined by is a continuous linear functional on H and so by the Riesz representation theorem, there exists a unique vector in H, denoted by A^* z, such that or equivalently, such that The assignment thus induces a function called the of A : H \to Z whose defining condition is The adjoint is necessarily a continuous (equivalently, a bounded) linear operator. If H is finite dimensional with the standard inner product and if M is the transformation matrix of A with respect to the standard orthonormal basis then M's conjugate transpose is the transformation matrix of the adjoint A^*.

Adjoints are transposes

It is also possible to define the or of which is the map defined by sending a continuous linear functionals to where the composition is always a continuous linear functional on H and it satisfies (this is true more generally, when H and Z are merely normed spaces). So for example, if z \in Z then {}^{t}A sends the continuous linear functional (defined on Z by ) to the continuous linear functional (defined on H by ); using bra-ket notation, this can be written as where the juxtaposition of with A on the right hand side denotes function composition: The adjoint is actually just to the transpose when the Riesz representation theorem is used to identify Z with Z^* and H with H^. Explicitly, the relationship between the adjoint and transpose is: which can be rewritten as: Alternatively, the value of the left and right hand sides of at any given z \in Z can be rewritten in terms of the inner products as: so that holds if and only if holds; but the equality on the right holds by definition of A^ z. The defining condition of A^* z can also be written if bra-ket notation is used.

Descriptions of self-adjoint, normal, and unitary operators

Assume Z = H and let Let A : H \to H be a continuous (that is, bounded) linear operator. Whether or not A : H \to H is self-adjoint, normal, or unitary depends entirely on whether or not A satisfies certain defining conditions related to its adjoint, which was shown by to essentially be just the transpose Because the transpose of A is a map between continuous linear functionals, these defining conditions can consequently be re-expressed entirely in terms of linear functionals, as the remainder of subsection will now describe in detail. The linear functionals that are involved are the simplest possible continuous linear functionals on H that can be defined entirely in terms of A, the inner product on H, and some given vector h \in H. Specifically, these are and where Self-adjoint operators A continuous linear operator A : H \to H is called self-adjoint if it is equal to its own adjoint; that is, if A = A^. Using, this happens if and only if: where this equality can be rewritten in the following two equivalent forms: Unraveling notation and definitions produces the following characterization of self-adjoint operators in terms of the aforementioned continuous linear functionals: A is self-adjoint if and only if for all z \in H, the linear functional is equal to the linear functional ; that is, if and only if where if bra-ket notation is used, this is Normal operators A continuous linear operator A : H \to H is called normal if which happens if and only if for all z, h \in H, Using and unraveling notation and definitions produces the following characterization of normal operators in terms of inner products of continuous linear functionals: A is a normal operator if and only if where the left hand side is also equal to The left hand side of this characterization involves only linear functionals of the form while the right hand side involves only linear functions of the form (defined as above ). So in plain English, characterization says that an operator is normal when the inner product of any two linear functions of the first form is equal to the inner product of their second form (using the same vectors z, h \in H for both forms). In other words, if it happens to be the case (and when A is injective or self-adjoint, it is) that the assignment of linear functionals is well-defined (or alternatively, if is well-defined) where h ranges over H, then A is a normal operator if and only if this assignment preserves the inner product on H^. The fact that every self-adjoint bounded linear operator is normal follows readily by direct substitution of A^* = A into either side of This same fact also follows immediately from the direct substitution of the equalities into either side of. Alternatively, for a complex Hilbert space, the continuous linear operator A is a normal operator if and only if for every z \in H, which happens if and only if Unitary operators An invertible bounded linear operator A : H \to H is said to be unitary if its inverse is its adjoint: By using, this is seen to be equivalent to Unraveling notation and definitions, it follows that A is unitary if and only if The fact that a bounded invertible linear operator A : H \to H is unitary if and only if (or equivalently, ) produces another (well-known) characterization: an invertible bounded linear map A is unitary if and only if Because A : H \to H is invertible (and so in particular a bijection), this is also true of the transpose This fact also allows the vector z \in H in the above characterizations to be replaced with A z or A^{-1} z, thereby producing many more equalities. Similarly, ,\cdot, can be replaced with A(\cdot) or

Citations