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 bijectiveisometry) 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.
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 antilinear 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 
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 antilinear functionals, all continuous antilinear functionals ) on
Mathematics vs. physics notations and definitions of inner productEdit
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 antilinear in its second coordinate; equivalently, the map is antilinear 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 opposite, meaning that the inner product is linear in the second coordinate and antilinear in the other coordinate.
This article will not choose one definition over the other.
Instead, the assumptions made above make it so that the mathematics notation satisfies the mathematical convention/definition for the inner product (that is, linear in the first coordinate and antilinear in the other), while the physics bra–ket notation satisfies the physics convention/definition for the inner product (that is, linear in the second coordinate and antilinear in the other). Consequently, the above two assumptions makes the notation used in each field consistent with that field's convention/definition for which coordinate is linear and which is antilinear.
Canonical norm and inner product on the dual space and anti-dual spaceEdit
The canonical norm on the (continuous) 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 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
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
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
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).
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
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
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 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).
(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
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
If the orthonormal basis is a sequence then this becomes
and if is written as then
Example in finite dimensions using matrix transformationsEdit
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 Hermitianpositive-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 spaceEdit
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
Canonical injections into the dual and anti-dualEdit
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:
which is a bijective antilinear isometry that satisfies
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
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
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
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).
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  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:
To show that fix
The definition of implies
so it remains to show that If then
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: