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 (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.
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 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, which is a map valued in H's underlying field which 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:
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 ⟨ y | • ⟩ = ⟨ •, y ⟩ : H → 𝔽 and defined by
h ↦ ⟨ y | h ⟩ = ⟨ h, y ⟩ for all
is a linear functional on
In fact, this linear functional is continuous, so ⟨ y | • ⟩ = ⟨ •, y ⟩ ∈ H*.
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 ⟨ • | y ⟩ = ⟨ y, • ⟩ : H → 𝔽 and defined by
h ↦ ⟨ h | y ⟩ = ⟨ y, h ⟩ for all
is an antilinear functional on H
In fact, this antilinear functional is continuous, so ⟨ • | y ⟩ = ⟨ y, • ⟩ ∈
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 ⟨ x | x ⟩ = ⟨ x, x ⟩ is a non-negative real number and the map
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 notions
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
The same equations that were used above can also be used to define a norm and inner product on 's anti-dual space
Theorem — Let be a Hilbert space whose inner product is linear in its first argument and antilinear in its second argument (the notation is used in physics). For every continuous linear functional there exists a unique such that
Importantly for complex Hilbert spaces, note that the vector is always located in the antilinear coordinate of the inner product (no matter which notation is used).[note 1]
Consequently, the map defined by is a bijective antilinear isometry whose inverse is the antilinear isometry defined by For the physics notation for the functional is the bra where explicitly this means that which complements the ket notation defined by
Let Then is closed subspace of If (or equivalently, if φ = 0) then we take and we're done. So assume
It is first shown that is one-dimensional. Using Zorn's lemma or the well-ordering theorem it can be shown that there exists some non-zero vector in — proving this is left as an exercise to the reader. We continue: Let and be nonzero vectors in Then and and there must exist a nonzero real number such that This implies that and so Since this implies that as desired.
Now let be a unit vector in For arbitrary let be the orthogonal projection of onto Then and (from the properties of orthogonal projections), so that and
Using the notation from the theorem above, we now provide ways of constructing from
If φ = 0 then fφ := 0 and otherwise for any
If is a unit vector then
If g is a unit vector satisfying the above condition then the same is true of -g (the only other unit vector in ). However, so both these vectors result in the same
If φ(x) ≠ 0 and xK is the orthogonal projection of onto ker φ, then [note 2]
Suppose φ ≠ 0 and let where note that since is real and is a proper subset of If we reinterpret as a real Hilbert space Hℝ (with the usual real-valued inner product defined by ), then has real codimension 1 in where has real codimension 1 in Hℝ, and (i.e. is perpendicular to with respect to ).
In the theorem and constructions above, if we replace with its real Hilbert space counterpart Hℝ and if we replace φ with Re φ then meaning that we will obtain the exact same vector by using (Hℝ, ⟨⋅, ⋅⟩ℝ) and the real linear functional Re φ as we did with the origin complex Hilbert space (H, ⟨⋅, ⋅⟩) and original complex linear functional φ (with identical norm values as well).
Given any continuous linear functional the corresponding element can be constructed uniquely by
where is an orthonormal basis of H, and the value of does not vary by choice of basis. Thus, if then
Canonical injection 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:
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 H:
φy = ⟨ y | • ⟩ = ⟨ •, y ⟩ : H → 𝔽 defined by h ↦ ⟨ y | h ⟩ = ⟨ h, y ⟩
This map is an element of which is the continuous dual space of
The canonical map from into its dual is the antilinear operator
defined by y ↦ φy = ⟨ • | y ⟩ = ⟨ y, • ⟩
which is also an injectiveisometry. 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 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:
⟨ • | y ⟩ = ⟨ y, • ⟩ : H → 𝔽 defined by h ↦ ⟨ h | y ⟩ = ⟨ y, h ⟩,
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 To make this explicit, let and be the bijective antilinear isometries defined respectively by
g ↦ ⟨ g | • ⟩H = ⟨ •, g ⟩H and z ↦ ⟨ z | • ⟩Z = ⟨ •, z ⟩Z
so that by definition
for all and for all
The relationship between the adjoint and transpose can be shown (see footnote for proof)[note 3] to be:
which can be rewritten as:
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
g ↦ ⟨ g | • ⟩H = ⟨ •, g ⟩H
so that by definition
Given a vector let denote the continuous linear functional ; that is, The resulting of plugging some given into the functional is the scalar where is the notation that is used instead of or
The assignment is just the isometric antilinear isomorphism so holds for all and all scalars
Given a continuous linear functional let denote the vector ; that is, The defining condition of the vector is the technically correct but unsightly equality
which is why the notation is used in place of The defining condition becomes
The assignment is just the isometric antilinear isomorphism so holds for all and all scalars
For any given vector the notation is used to denote ; that is, The notation and is used in place of and respectively. As expected, and really is just the scalar
The inverse map of can be described as follows. Given a non-zero element of H*, the orthogonal complement of the kernel of is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set Then =
Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907 (see references).
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.
^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 0 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