Riesz representation theorem

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 isomorphism is a particular natural isomorphism.

Preliminaries and notation

Let be a Hilbert space over a field where is either the real numbers or the complex numbers If then is called a . Every real Hilbert space can be extended to be a dense subset of a unique 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 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 a complex Hilbert space, or a real complex Hilbert space.

Linear and antilinear maps

By definition, an [Antilinear map|] is a map between vector spaces that is :
and :
where is the conjugate of the complex number, given by.
In contrast, a map is linear if it is additive and [Homogeneous function|]:
Every constant 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 is continuous if and only if it is bounded; the same is true of antilinear maps. The inverse of any antilinear bijection is again an antilinear bijection. The composition of two linear maps is a map.
Continuous dual and anti-dual spaces
A on is a function whose codomain is the underlying scalar field
Denote by functionals on which is called the of
If then linear functionals on 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 has an associated inner product valued in 's underlying scalar field that is linear in one coordinate and antilinear in the other.
If 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, so there can be no such confusion.
In mathematics, the inner product on a Hilbert space 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:

The map is linear in its first coordinate; equivalently, the map is linear in its second coordinate. That is, for fixed the map
with
is a linear functional on This linear functional is continuous, so
The map is antilinear in its coordinate; equivalently, the map is antilinear in its coordinate. That is, for fixed the map
with
is an antilinear functional on This antilinear functional is continuous, so

In computations, one must consistently use either the mathematics notation, which is ; or the physics notation, which is.

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

If then is a non-negative real number and the map
defines a canonical norm on that makes into a normed space.
As with all normed spaces, the dual space carries a canonical norm, called the, that is defined by
The canonical norm on the anti-dual space denoted by is defined by using this same equation:
This canonical norm on 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 into a Hilbert space. There are now two ways of defining a norm on the norm induced by this inner product and the usual dual norm. These norms are the same; explicitly, this means that the following holds for every
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
The same equations that were used above can also be used to define a norm and inner product on 's anti-dual space
Canonical isometry between the dual and antidual
The complex conjugate of a functional which was defined above, satisfies
for every 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 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 and are if which happens if and only if for all scalars The orthogonal complement of a subset is
which is always a closed vector subspace of
The Hilbert projection theorem guarantees that for any nonempty closed convex subset of a Hilbert space there exists a unique vector such that that is, is the global minimum point of the function defined by

Statement

Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907.
Let denote the underlying scalar field of
Fix
Define by which is a linear functional on since is in the linear argument.
By the Cauchy–Schwarz inequality,
which shows that is bounded and that
It remains to show that
By using in place of it follows that
.
Thus that
The proof above did not use the fact that is complete, which shows that the formula for the norm holds more generally for all inner product spaces.
Suppose are such that and for all
Then
which shows that is the constant linear functional.
Consequently which implies that
Let
If then taking completes the proof so assume that and
The continuity of implies that is a closed subspace of .
Let
denote the orthogonal complement of in
Because is closed and is a Hilbert space, can be written as the direct sum .
Because there exists some non-zero
For any
which shows that where now implies
Solving for shows that
which proves that the vector satisfies
Applying the norm formula that was proved above with shows that
Also, the vector has norm and satisfies
It can now be deduced that is -dimensional when
Let be any non-zero vector. Replacing with in the proof above shows that the vector satisfies for every The uniqueness of the vector representing implies that which in turn implies that and Thus every vector in 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 is often interpreted geometrically by identifying it with the affine hyperplane . In particular, the norm of should somehow be interpretable as the "norm of the hyperplane ". When then the Riesz representation theorem provides such an interpretation of 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 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 entirely in terms of its associated affine hyperplane . Defining the infimum formula
will also hold when
When the supremum is taken in, then the supremum of the empty set is but if the supremum is taken in the non-negative reals then this supremum is instead in which case the supremum formula will also hold when .

Constructions of the representing vector

Using the notation from the theorem above, several ways of constructing from are now described.
If then ; in other words,
This special case of 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 then
.
If is a unit vector satisfying the above condition then the same is true of which is also a unit vector in However, so both these vectors result in the same
Orthogonal projection onto kernel
If is such that and if is the orthogonal projection of onto then
Orthonormal basis
Given an orthonormal basis of and a continuous linear functional the vector can be constructed uniquely by
where all but at most countably many will be equal to and where the value of does not actually depend on choice of orthonormal basis.
If is written as then
and
If the orthonormal basis is a sequence then this becomes
and if is written as then

Example in finite dimensions using matrix transformations

Consider the special case of with the standard inner product
where are represented as column matrices and with respect to the standard orthonormal basis on and where denotes the conjugate transpose of
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 is the vector
To see why, identify every vector in with the column matrix
so that is identified with
As usual, also identify the linear functional with its transformation matrix, which is the row matrix so that and the function is the assignment where the right hand side is matrix multiplication. Then for all
which shows that satisfies the defining condition of the Riesz representation of
The bijective antilinear isometry defined in the corollary to the Riesz representation theorem is the assignment that sends to the linear functional on defined by
where under the identification of vectors in with column matrices and vector in with row matrices, is just the assignment
As described in the corollary, 's inverse is the antilinear isometry which was just shown above to be:
where in terms of matrices, is the assignment
Thus in terms of matrices, each of and is just the operation of conjugate transposition matrices then is identified with the space of all row.
This example used the standard inner product, which is the map but if a different inner product is used, such as where 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 is a complex Hilbert space with inner product
When the Hilbert space is reinterpreted as a real Hilbert space then it will be denoted by where the inner-product on is the real part of 's inner product; that is:
The norm on induced by is equal to the original norm on and the continuous dual space of is the set of all -valued bounded -linear functionals on .
Let and denote the real and imaginary parts of a linear functional so that
The formula expressing a linear functional in terms of its real part is
where for all
It follows that and that if and only if
It can also be shown that where and are the usual operator norms.
In particular, a linear functional is bounded if and only if its real part is bounded.
Representing a functional and its real part
The Riesz representation of a continuous linear function 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 obtained in so it is the unique vector that satisfies for all
The real part of is a continuous real linear functional on 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 that satisfies for all
The conclusion is
This follows from the main theorem because and if 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 is replaced with its real Hilbert space counterpart and if is replaced with then This means that vector 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 .
Furthermore, if then is perpendicular to with respect to where the kernel of is be a proper subspace of the kernel of its real part Assume now that
Then because and is a proper subset of The vector subspace has real codimension in while has codimension in and That is, 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 into the coordinate of the inner product and letting the variable vary over the coordinate results in an functional:
This map is an element of which is the continuous anti-dual space of
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. Consequently, every antilinear functional on can be written 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 let denote the continuous linear functional ; 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 and all scalars
The result of plugging some given 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
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 the notation is used to denote ; that is,
The assignment is just the identity map which is why holds for all and all scalars
The notation and is used in place of and respectively. As expected, and really is just the scalar

Adjoints and transposes

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

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 and it satisfies .
So for example, if then sends the continuous linear functional to the continuous linear functional ;
using bra-ket notation, this can be written as where the juxtaposition of with 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 with and with
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 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
The defining condition of can also be written
if bra-ket notation is used.

Descriptions of self-adjoint, normal, and unitary operators

Assume and let
Let be a continuous linear operator.
Whether or not is self-adjoint, normal, or unitary depends entirely on whether or not satisfies certain defining conditions related to its adjoint, which was shown by to essentially be just the transpose
Because the transpose of 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 that can be defined entirely in terms of the inner product on and some given vector
Specifically, these are and where
Self-adjoint operators
A continuous linear operator is called self-adjoint if it is equal to its own adjoint; that is, if 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: is self-adjoint if and only if for all 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 is called normal if which happens if and only if for all
Using and unraveling notation and definitions produces the following characterization of normal operators in terms of inner products of continuous linear functionals: 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 .
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.
In other words, if it happens to be the case that the assignment of linear functionals is well-defined where ranges over then is a normal operator if and only if this assignment preserves the inner product on
The fact that every self-adjoint bounded linear operator is normal follows readily by direct substitution of 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 is a normal operator if and only if for every which happens if and only if
Unitary operators
An invertible bounded linear operator 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 is unitary if and only if
The fact that a bounded invertible linear operator is unitary if and only if produces another characterization: an invertible bounded linear map is unitary if and only if
Because is invertible, this is also true of the transpose This fact also allows the vector in the above characterizations to be replaced with or thereby producing many more equalities. Similarly, can be replaced with or