Riesz representation theorem

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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[edit]

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[edit]

By definition, an antilinear map (also called a conjugate-linear map) is a map between vector spaces that is additive:

     for all

and antilinear (also called conjugate-linear or conjugate-homogeneous):

     for all and all scalar

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

     for all and all scalar

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 [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 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[edit]

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:

     for all
Completing definitions of the inner product

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

  1. 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*.
  2. 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[edit]

If then x | x = ⟨ x, x is a non-negative real number and the map

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

     for every

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

     for every

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 [1]

Canonical isometry between the dual and antidual

The complex conjugate of a functional which was defined above, satisfies

     and,     

for every and every This says exactly that that the canonical antilinear bijection defined by

     where     

as well as its inverse are antilinear isometries and consequently also homeomorphisms. If then and this canonical map reduces down to the identity map.

Riesz representation theorem[edit]

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

     for all

and moreover,

  • 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

Proof —

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 Thus

Because of this, we take We also see that

From the Cauchy-Bunyakovsky-Schwarz inequality and so if has unit norm then Since has unit norm, we have

Observations:

  • So in particular, we always have is real, where fφ = 0 φ = 0.
  • 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.

Constructions[edit]

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 H (with the usual real-valued inner product defined by ), then has real codimension 1 in where has real codimension 1 in H, and (i.e. is perpendicular to with respect to ).
    • In the theorem and constructions above, if we replace with its real Hilbert space counterpart H and if we replace φ with Re φ then meaning that we will obtain the exact same vector by using (H, ⟨⋅, ⋅⟩) 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[edit]

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       hy | h = ⟨ h, y
    This map is an element of which is the continuous dual space of The canonical map from into its dual [1] is the antilinear operator
          defined by       y ↦ φy = ⟨ • | y = ⟨ y, • ⟩
    which is also an injective isometry.[1] 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.[1]
  • 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       hh | y = ⟨ y, h,
    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
          defined by       y⟨ • | y = ⟨ y, • ⟩
    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 antilinear bijective isometry that was defined above, then the following equality holds:

Adjoints and transposes[edit]

Let be a continuous linear operator between Hilbert spaces and As before, let and The adjoint of is the linear operator defined by the condition:

     for all and all

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

gg | • ⟩H = ⟨ •, gH      and      zz | • ⟩Z = ⟨ •, zZ

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:

     and     

Extending the bra-ket notation to bras and kets[edit]

Let be a Hilbert space and as before, let Let be the bijective antilinear isometry defined by

gg | • ⟩H = ⟨ •, gH

so that by definition

     for all
Bras

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

     for all

which is why the notation is used in place of The defining condition becomes

     for all

The assignment is just the isometric antilinear isomorphism so holds for all and all scalars

Kets

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

Properties of induced antilinear map[edit]

The mapping : HH* 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 λ.
  • If the base field is then for all complex numbers λ, where denotes the complex conjugation of

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 =

Alternatively, the assignment can 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 H*.

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.

See also[edit]

Notes[edit]

  1. ^ a b c d e f g h i j k l Trèves 2006, pp. 112-123.
  1. ^ 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
  2. ^ Since we must have Now use and and solve for
  3. ^ To show that fix The definition of implies so it remains to show that If then as desired. ◼

References[edit]

  • Fréchet, M. (1907). "Sur les ensembles de fonctions et les opérations linéaires". Les Comptes rendus de l'Académie des sciences (in French). 144: 1414–1416.
  • P. Halmos Measure Theory, D. van Nostrand and Co., 1950.
  • P. Halmos, A Hilbert Space Problem Book, Springer, New York 1982 (problem 3 contains version for vector spaces with coordinate systems).
  • Riesz, F. (1907). "Sur une espèce de géométrie analytique des systèmes de fonctions sommables". Comptes rendus de l'Académie des Sciences (in French). 144: 1409–1411.
  • Riesz, F. (1909). "Sur les opérations fonctionnelles linéaires". Comptes rendus de l'Académie des Sciences (in French). 149: 974–977.
  • Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
  • Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.