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.

Let be a Hilbert space over a field where is either the real numbers or the complex numbers If (resp. if ) then is called a *complex Hilbert space* (resp. a *real Hilbert space*). 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 *or* complex Hilbert space.

By definition, an *antilinear map* (also called a *conjugate-linear map*) is a map between vector spaces that is *additive*:
and *antilinear* (also called *conjugate-linear* or *conjugate-homogeneous*):
where is the conjugate of the complex number , given by .

In contrast, a map is linear if it is additive and *homogeneous*:

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 (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 *anti*linear maps is a *linear* map.

**Continuous dual and anti-dual spaces**

A *functional* on is a function whose codomain is the underlying scalar field
Denote by (resp. by the set of all continuous linear (resp. continuous antilinear) functionals on which is called the *(continuous) dual space* (resp. the *(continuous) anti-dual space*) of ^{[1]}
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 *conjugate of * 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

- all functionals (resp. all linear functionals, all continuous linear functionals ) on

onto the set of

- all functionals (resp. all
*anti*linear functionals, all continuous*anti*linear functionals ) on

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 (as described in detail below).
If is a complex Hilbert space (meaning, if ), which is very often the case, then which coordinate is antilinear and which is linear becomes a *very* important technicality.
However, if then the inner product is a symmetric map that is simultaneously linear in each coordinate (that is, bilinear) and antilinear in each coordinate. Consequently, the question of which coordinate is linear and which is antilinear is irrelevant for real Hilbert spaces.

**Notation for the inner product**

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 instead. In this article, these two notations will be related by the equality:

**Competing definitions of the inner product**

The maps and are assumed to have the following two properties:

- The map is
*linear*in its*first*coordinate; equivalently, the map is linear in its*second*coordinate. Explicitly, this means that for every fixed the map that is denoted by and defined by is a linear functional on- In fact, this linear functional is continuous, so

- The map is
*anti*linear in its*second*coordinate; equivalently, the map is*anti*linear in its*first*coordinate. Explicitly, this means that for every fixed the map that is denoted by and defined by is an antilinear functional on- In fact, this antilinear functional is continuous, so

In mathematics, the prevailing convention (i.e. the definition of an inner product) is that the inner product is *linear in the first* coordinate and antilinear in the other coordinate. In physics, the convention/definition is unfortunately the

If then is a non-negative real number and the map

defines a canonical norm on that makes into a normed space.^{[1]}
As with all normed spaces, the (continuous) dual space carries a canonical norm, called the *dual norm*, that is defined by^{[1]}

The canonical norm on the (continuous) anti-dual space denoted by is defined by using this same equation:^{[1]}

This canonical norm on satisfies the parallelogram law, which means that the polarization identity can be used to define a *canonical inner product on * 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 (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

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 ^{[1]}

**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.

Two vectors and are *orthogonal* if which happens if and only if for all scalars ^{[2]} 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 (unique) global minimum point of the function defined by

**Riesz representation theorem** — Let be a Hilbert space whose inner product is linear in its *first* argument and antilinear in its second argument and let be the corresponding physics notation. For every continuous linear functional there exists a unique vector called the *Riesz representation of * such that^{[3]}

Importantly for *complex* Hilbert spaces, is always located in the *antilinear* coordinate of the inner product.^{[note 1]}

Furthermore, the length of the representation vector is equal to the norm of the functional: and is the unique vector with It is also the unique element of minimum norm in ; that is to say, is the unique element of satisfying Moreover, any non-zero can be written as

**Corollary** — The *canonical map from into its dual* ^{[1]} is the injective *anti*linear operator isometry^{[note 2]}^{[1]}
The Riesz representation theorem states that this map is surjective (and thus bijective) when is complete and that its inverse is the bijective isometric antilinear isomorphism
Consequently, *every* continuous linear functional on the Hilbert space can be written uniquely in the form ^{[1]} where for every
The assignment can also be viewed as a bijective *linear* isometry into the anti-dual space of ^{[1]} which is the complex conjugate vector space of the continuous dual space

The inner products on and are related by and similarly,

The set satisfies and so when then can be interpreted as being the affine hyperplane^{[note 3]} that is parallel to the vector subspace and contains

For the physics notation for the functional is the bra where explicitly this means that which complements the ket notation defined by In the mathematical treatment of quantum mechanics, the theorem can be seen as a justification for the popular bra–ket notation. The theorem says that, every bra has a corresponding ket and the latter is unique.

Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907 (see references).

Proof
^{[4]} |
---|

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 (equivalently, continuous) and that It remains to show that By using in place of it follows that (the equality holds because is real and non-negative). 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 (or equivalently, if ) then taking completes the proof so assume that and
The continuity of implies that is a closed subspace of (because and is a closed subset of ).
Let
denote the orthogonal complement of in
Because is closed and is a Hilbert space, 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 (non-zero) 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. |

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 (the kernel is also often visualized alongside although knowing is enough to reconstruct because if then and otherwise ). 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^{[note 3]}
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 (because with this formula, knowing only the *set* is enough to describe the norm of its associated linear *functional*). Defining the infimum formula
will also hold when
When the supremum is taken in (as is typically assumed), then the supremum of the empty set is but if the supremum is taken in the non-negative reals (which is the image/range of the norm when ) then this supremum is instead in which case the supremum formula will also hold when (although the atypical equality is usually unexpected and so risks causing confusion).

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 (meaning ) then (this is true even if because in this case ). 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^{[proof 1]}

**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 (that is, using any other orthonormal basis for will result in the same vector). If is written as then and

If the orthonormal basis is a sequence then this becomes and if is written as then

Consider the special case of (where is an integer) with the standard inner product
where are represented as column matrices and with respect to the standard orthonormal basis on (here, is at its ^{th} coordinate and everywhere else; as usual, will now be associated with the dual basis) 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 (although between different spaces of matrices: if is identified with the space of all column (respectively, row) matrices then 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 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.

See also: Complexification |

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 (real) 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 *real*-valued bounded -linear functionals on (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 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 (with identical norm values as well).

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 *real* codimension in and That is, is perpendicular to with respect to

**Induced linear map into anti-dual**

The map defined by placing into the *linear* coordinate of the inner product and letting the variable vary over the *antilinear* coordinate results in an *antilinear* functional:

This map is an element of which is the continuous anti-dual space of
The *canonical map from into its anti-dual* ^{[1]} is the *linear* operator
which is also an injective isometry.^{[1]}
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 can be written (uniquely) in this form.^{[1]}

If is the canonical *anti*linear bijective isometry that was defined above, then the following equality holds:

Main article: Bra–ket notation |

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 ^{[note 6]}

**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

Let be a continuous linear operator between Hilbert spaces and As before, let and

Denote by the usual bijective antilinear isometries that satisfy:

Main articles: Hermitian adjoint and Conjugate transpose |

For every the scalar-valued map ^{[note 7]} 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 *adjoint* of whose defining condition is
The adjoint is necessarily a continuous (equivalently, a bounded) 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

Main article: Transpose of a linear map |

See also: Transpose |

It is also possible to define the *transpose* or *algebraic adjoint* 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 (this is true more generally, when and are merely normed spaces).^{[5]}
So for example, if then sends the continuous linear functional (defined on by ) to the continuous linear functional (defined on by );^{[note 7]}
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 ^{[2]} when the Riesz representation theorem is used to identify with and with

Explicitly, the relationship between the adjoint and transpose is:

(Adjoint-transpose) |

which can be rewritten as:

To show that fix The definition of implies so it remains to show that If then as desired.

Alternatively, the value of the left and right hand sides of (**Adjoint-transpose**) 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.

Assume and let Let be a continuous (that is, bounded) 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 (**Adjoint-transpose**) 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 ^{[note 7]} where

**Self-adjoint operators**

See also: Self-adjoint operator, Hermitian matrix, and Symmetric matrix |

A continuous linear operator is called self-adjoint if it is equal to its own adjoint; that is, if Using (**Adjoint-transpose**), 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 ^{[note 7]} is equal to the linear functional ; that is, if and only if

(Self-adjointness functionals) |

where if bra-ket notation is used, this is

**Normal operators**

See also: Normal operator and Normal matrix |

A continuous linear operator is called normal if which happens if and only if for all

Using (**Adjoint-transpose**) and unraveling notation and definitions produces^{[proof 2]} the following characterization of normal operators in terms of inner products of continuous linear functionals: is a normal operator if and only if

(Normality functionals) |

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^{[note 7]}).
So in plain English, characterization (**Normality functionals**) 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 for both forms).
In other words, if it happens to be the case (and when is injective or self-adjoint, it is) that the assignment of linear functionals is well-defined (or alternatively, if 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 (**Self-adjointness functionals**) into either side of (**Normality functionals**).

Alternatively, for a complex Hilbert space, the continuous linear operator is a normal operator if and only if for every ^{[2]} which happens if and only if

**Unitary operators**

See also: Unitary transformation and Unitary matrix |

An invertible bounded linear operator is said to be unitary if its inverse is its adjoint:
By using (**Adjoint-transpose**), 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 (or equivalently, ) produces another (well-known) characterization: an invertible bounded linear map is unitary if and only if

Because is invertible (and so in particular a bijection), 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