- This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.
Riesz representation theorem, sometimes called Riesz–Fréchet representation theorem, named 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 one as will be described next; a natural isomorphism.
Preliminaries and notation
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 (i.e. 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.
Linear and antilinear maps
- for all
and antilinear (also called conjugate-linear or conjugate-homogeneous):
- for all and all scalar
- for all and all scalar
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 f is the functional denoted by
- and defined by
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 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 (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 a symmetric map that is simultaneously linear in each coordinate (i.e. 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:
- for all
- Completing 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
- for all
- 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
- for all
- 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 chose 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 (i.e. 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 (i.e. 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 space
If then is a non-negative real number and the map
- for every
- for every
where this inner product turns into a Hilbert space. Moreover, the canonical norm induced by this inner product (i.e. the norm defined by ) is consistent with the dual norm (i.e. as defined above by the supremum over the unit ball); 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
- 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
Riesz representation theorem
Let denote the underlying scalar field.
Let Then is closed subspace of If (or equivalently, if ) then taking completes the proof so assume that and
It is first shown that the orthogonal complement of with respect to 's inner product is a -dimensional vector space over It can be shown using Zorn's lemma or the well-ordering theorem that that is, there exists some non-zero vector in The Hilbert projection theorem can also be used. It is now shown that has dimension at most over Let nonzero vectors. Then and are non-zero scalars and so there must exist a scalar such that (specifically, ). This implies that and consequently, Because is a vector space, Since this implies that as desired.
Now let be a unit vector. For arbitrary let be the orthogonal projection of onto Then and (from the properties of orthogonal projections). Consequently, while implies Thus
Letting it follows that is equal to the map
As is now shown, also holds, where the left hand side is Because so it remains to prove the reverse inequality. The Cauchy-Bunyakovsky-Schwarz inequality implies that
Because was arbitrary, it follows that as desired.
The formulas for the inner products follow from the polarization identity.
- So in particular, is always real, where if and only if if and only if
- Showing that there is a non-zero vector in relies on the continuity of and the Cauchy completeness of This is the only place in the proof in which these properties are used.
- A non-trivial continuous linear functional is often interpreted geometrically by identifying it with the affine hyperplane (the kernel is also often visualized although knowing is enough 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 as follows: Because so implies and thus This can also be seen applying the Hilbert projection theorem to and concluding that the minimizer of 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 formula will also hold when
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
- 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
- If is such that and if is the orthogonal projection of onto then[proof 1]
- Given any continuous linear functional the corresponding element can be constructed uniquely by
where is an orthonormal basis of and the value of does not vary by choice of basis. Thus, if then
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 (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
- Representing a functional and its real part
Let and as usual, let be such that for all Let
Assume now that Then because and is a proper subset of The kernel has real codimension in where has real codimension in and That is, is perpendicular to with respect to
Properties of canonical injections from a Hilbert space to its dual and anti-dual
For every the inner product on can be used to define two continuous (i.e. bounded) canonical maps.
- Induced antilinear map into dual
The map defined by placing into the antilinear coordinate of the inner product and letting the variable vary over the linear coordinate results in a linear functional on :
- defined by
- �� defined by
which is also an injective isometry. The Riesz representation theorem states that this map is surjective (and thus bijective). Consequently, every continuous linear functional on can be written (uniquely) in this form.
The mapping defined by = is an isometric antilinear isomorphism, meaning that:
- is bijective.
- The norms of and agree:
- Using this fact, this map could be used to give an equivalent definition of the canonical dual norm of The canonical inner product on could be defined similarly.
- is additive:
- If the base field is then for all real numbers
- If the base field is then for all complex numbers where denotes the complex conjugation of
The inverse map of can be described as follows. Given a non-zero element of the orthogonal complement of the kernel of is a one-dimensional subspace of Take a non-zero element in that subspace, and set Then
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.
- 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:
- defined by
- defined by
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 can be written (uniquely) in this form.
Extending the bra-ket notation to bras and kets
Let be a Hilbert space and as before, let Let be the bijective antilinear isometry defined by
so that by definition
- for all
Given a vector let denote the continuous linear functional ; that is,
which 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 resulting of plugging some given into the functional is the scalar which may be denoted by [note 2]
- 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
- for all
which is why the notation is used in place of The defining condition becomes
- for all
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 and be the bijective antilinear isometries defined respectively by
Definition of the adjoint
For every the scalar-valued map [note 3] 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 (i.e. bounded) linear operator.
Adjoints are transposes
It is also possible to define the transpose of which is the map defined by sending a continuous linear functionals to
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 can be shown[proof 2] to be:
which can be rewritten as:
Given any the left and right hand sides of equality (Adjoint-transpose) can be rewritten in terms of the inner products:
Descriptions of self-adjoint, normal, and unitary operators
Assume and let Let be a continuous (i.e. 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 These "elementary -induced" continuous linear functionals are and [note 3] where
- Self-adjoint operators
Unraveling notation and definitions produces the following characterization of self-adjoint operators in terms of the aforementioned "elementary -induced" continuous linear functionals: is self-adjoint if and only if for all the linear functional [note 3] is equal to the linear functional ; that is, if and only if
- Normal operators
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 3] the following characterization of normal operators in terms of inner products of the "elementary -induced" continuous linear functionals: is a normal operator if and only if
The left hand side of this characterization is also equal to The continuous linear functionals and are defined as above.[note 3]
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  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 (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
- If then the inner product will be symmetric so it doesn't matter which coordinate of the inner product the element is placed into because the same map will result. But if then except for the constant map, antilinear functionals on are completely distinct from linear functionals on which makes the coordinate that is placed into is very important. For a non-zero to induce a linear functional (rather than an antilinear functional), must be placed into the antilinear coordinate of the inner product. If it is incorrectly placed into the linear coordinate instead of the antilinear coordinate then the resulting map will be the antilinear map which is not a linear functional on and so it will not be an element of the continuous dual space
- The usual notation for plugging an element into a linear map is and sometimes Replacing with produces or which is unsightly (despite being consistent with the usual notation used with functions). Consequently, the symbol is appended to the end, so that the notation is used instead to denote this value
- The notation denotes the continuous linear functional defined by
- This is because Now use and and solve for
- To show that fix The definition of implies so it remains to show that If then as desired.
- where and By definition of the adjoint, so taking the complex conjugate of both sides proves that From it follows that where and
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