Riesz representation theorem

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 ${\displaystyle H}$ be a Hilbert space over a field ${\displaystyle \mathbb {F} ,}$ where ${\displaystyle \mathbb {F} }$ is either the real numbers ${\displaystyle \mathbb {R} }$ or the complex numbers ${\displaystyle \mathbb {C} .}$ If ${\displaystyle \mathbb {F} =\mathbb {C} }$ (resp. if ${\displaystyle \mathbb {F} =\mathbb {R} }$) then ${\displaystyle H}$ 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 ${\displaystyle \mathbb {F} =\mathbb {R} }$) 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

By definition, an antilinear map (also called a conjugate-linear map) ${\displaystyle f:H\to Y}$ is a map between vector spaces that is additive:

${\displaystyle f(x+y)=f(x)+f(y)}$      for all ${\displaystyle x,y\in H,}$

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

${\displaystyle f(cx)={\overline {c}}f(x)}$      for all ${\displaystyle x\in H}$ and all scalar ${\displaystyle c\in \mathbb {F} .}$

In contrast, a map ${\displaystyle f:H\to Y}$ is linear if it is additive and homogeneous:

${\displaystyle f(cx)=cf(x)}$      for all ${\displaystyle x\in H}$ and all scalar ${\displaystyle c\in \mathbb {F} .}$

Every constant 0 map is always both linear and antilinear. If ${\displaystyle \mathbb {F} =\mathbb {R} }$ 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 ${\displaystyle H}$ is a function ${\displaystyle H\to \mathbb {F} }$ whose codomain is the underlying scalar field ${\displaystyle \mathbb {F} .}$ Denote by ${\displaystyle H^{*}}$ (resp. by ${\displaystyle {\overline {H}}^{*})}$ the set of all continuous linear (resp. continuous antilinear) functionals on ${\displaystyle H,}$ which is called the (continuous) dual space (resp. the (continuous) anti-dual space) of ${\displaystyle H.}$[1] If ${\displaystyle \mathbb {F} =\mathbb {R} }$ then linear functionals on ${\displaystyle H}$ are the same as antilinear functionals and consequently, the same is true for such continuous maps: that is, ${\displaystyle H^{*}={\overline {H}}^{*}.}$

One-to-one correspondence between linear and antilinear functionals

Given any functional ${\displaystyle f~:~H\to \mathbb {F} ,}$ the conjugate of f is the functional denoted by

${\displaystyle {\overline {f}}~:~H\to \mathbb {F} }$      and defined by      ${\displaystyle h\mapsto {\overline {f(h)}}.}$

This assignment is most useful when ${\displaystyle \mathbb {F} =\mathbb {C} }$ because if ${\displaystyle \mathbb {F} =\mathbb {R} }$ then ${\displaystyle f={\overline {f}}}$ and the assignment ${\displaystyle f\mapsto {\overline {f}}}$ reduces down to the identity map.

The assignment ${\displaystyle f\mapsto {\overline {f}}}$ defines an antilinear bijective correspondence from the set of

all functionals (resp. all linear functionals, all continuous linear functionals ${\displaystyle H^{*}}$) on ${\displaystyle H,}$

onto the set of

all functionals (resp. all antilinear functionals, all continuous antilinear functionals ${\displaystyle {\overline {H}}^{*}}$) on ${\displaystyle H.}$

Mathematics vs. physics notations and definitions of inner product

The Hilbert space ${\displaystyle H}$ has an associated inner product, which is a map ${\displaystyle H\times H\to \mathbb {F} }$ valued in H's underlying field ${\displaystyle \mathbb {F} ,}$ which is linear in one coordinate and antilinear in the other (as described in detail below). If ${\displaystyle H}$ is a complex Hilbert space (meaning, if ${\displaystyle \mathbb {F} =\mathbb {C} }$), which is very often the case, then which coordinate is antilinear and which is linear becomes a very important technicality. However, if ${\displaystyle \mathbb {F} =\mathbb {R} }$ 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 ${\displaystyle H}$ is often denoted by ${\displaystyle \left\langle \cdot ,\cdot \right\rangle }$ or ${\displaystyle \left\langle \cdot ,\cdot \right\rangle _{H}}$ while in physics, the bra-ket notation ${\displaystyle \left\langle \cdot |\cdot \right\rangle }$ or ${\displaystyle \left\langle \cdot |\cdot \right\rangle _{H}}$ is typically used instead. In this article, these two notations will be related by the equality:

${\displaystyle \left\langle x,y\right\rangle :=\left\langle y|x\right\rangle }$      for all ${\displaystyle x,y\in H.}$
Completing definitions of the inner product

The maps ${\displaystyle \left\langle \cdot ,\cdot \right\rangle }$ and ${\displaystyle \left\langle \cdot |\cdot \right\rangle }$ are assumed to have the following two properties:

1. The map ${\displaystyle \left\langle \cdot ,\cdot \right\rangle }$ is linear in its first coordinate; equivalently, the map ${\displaystyle \left\langle \cdot |\cdot \right\rangle }$ is linear in its second coordinate. Explicitly, this means that for every fixed ${\displaystyle y\in H,}$ the map that is denoted by y | • ⟩ = ⟨ •, y ⟩ : H → 𝔽 and defined by
h   ↦   y | h = ⟨ h, y      for all ${\displaystyle h\in H}$
is a linear functional on ${\displaystyle H.}$
• In fact, this linear functional is continuous, so y | • ⟩ = ⟨ •, y ⟩ ∈ H*.
2. The map ${\displaystyle \left\langle \cdot ,\cdot \right\rangle }$ is antilinear in its second coordinate; equivalently, the map ${\displaystyle \left\langle \cdot |\cdot \right\rangle }$ is antilinear in its first coordinate. Explicitly, this means that for every fixed ${\displaystyle y\in H,}$ the map that is denoted by ⟨ • | y = ⟨ y, • ⟩ : H → 𝔽 and defined by
h   ↦   h | y = ⟨ y, h      for all ${\displaystyle h\in H}$
is an antilinear functional on H
• In fact, this antilinear functional is continuous, so ⟨ • | y = ⟨ y, • ⟩ ∈ ${\displaystyle {\overline {H}}^{*}.}$

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 ${\displaystyle \left\langle \cdot ,\cdot \right\rangle }$ 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 ${\displaystyle \left\langle \cdot |\cdot \right\rangle }$ 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 ${\displaystyle x=y}$ then x | x = ⟨ x, x is a non-negative real number and the map

${\displaystyle \left\|x\right\|:={\sqrt {\left\langle x,x\right\rangle }}={\sqrt {\left\langle x|x\right\rangle }}}$

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

${\displaystyle \|f\|_{H^{*}}~:=~\sup _{\|x\|\leq 1,x\in H}|f(x)|\,}$      for every ${\displaystyle f\in H^{*}.}$

The canonical norm on the (continuous) anti-dual space ${\displaystyle {\overline {H}}^{*},}$ denoted by ${\displaystyle \|f\|_{{\overline {H}}^{*}},}$ is defined by using this same equation:[1]

${\displaystyle \|f\|_{{\overline {H}}^{*}}~:=~\sup _{\|x\|\leq 1,x\in H}|f(x)|\,}$      for every ${\displaystyle f\in {\overline {H}}^{*}.}$

This canonical norm on ${\displaystyle H^{*}}$ satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on ${\displaystyle H^{*},}$ which this article will denote by the notions

${\displaystyle \left\langle f,g\right\rangle _{H^{*}}:=\left\langle g|f\right\rangle _{H^{*}},}$

where this inner product turns ${\displaystyle H^{*}}$ into a Hilbert space. Moreover, the canonical norm induced by this inner product (i.e. the norm defined by ${\displaystyle f\mapsto {\sqrt {\left\langle f,f\right\rangle _{H^{*}}}}}$) 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 ${\displaystyle f\in H^{*}}$:

${\displaystyle \sup _{\|x\|\leq 1,x\in H}|f(x)|=\|f\|_{H^{*}}~=~{\sqrt {\langle f,f\rangle _{H^{*}}}}~=~{\sqrt {\langle f|f\rangle _{H^{*}}}}.}$

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 ${\displaystyle H^{*}.}$

The same equations that were used above can also be used to define a norm and inner product on ${\displaystyle H}$'s anti-dual space ${\displaystyle {\overline {H}}^{*}.}$[1]

Canonical isometry between the dual and antidual

The complex conjugate ${\displaystyle {\overline {f}}}$ of a functional ${\displaystyle f,}$ which was defined above, satisfies

${\displaystyle \left\|f\right\|_{H^{*}}~=~\left\|{\overline {f}}\right\|_{{\overline {H}}^{*}}}$      and,      ${\displaystyle \left\|{\overline {g}}\right\|_{H^{*}}~=~\left\|g\right\|_{{\overline {H}}^{*}}}$

for every ${\displaystyle f\in H^{*}}$ and every ${\displaystyle g\in {\overline {H}}^{*}.}$ This says exactly that that the canonical antilinear bijection defined by

${\displaystyle \operatorname {Cong} ~:~H^{*}\to {\overline {H}}^{*}}$      where      ${\displaystyle \operatorname {Cong} (f):={\overline {f}}}$

as well as its inverse ${\displaystyle \operatorname {Cong} ^{-1}~:~{\overline {H}}^{*}\to H^{*}}$ are antilinear isometries and consequently also homeomorphisms. If ${\displaystyle \mathbb {F} =\mathbb {R} }$ then ${\displaystyle H^{*}={\overline {H}}^{*}}$ and this canonical map ${\displaystyle \operatorname {Cong} :H^{*}\to {\overline {H}}^{*}}$ reduces down to the identity map.

Riesz representation theorem

Theorem — Let ${\displaystyle H}$ be a Hilbert space whose inner product ${\displaystyle \left\langle x,y\right\rangle }$ is linear in its first argument and antilinear in its second argument (the notation ${\displaystyle \left\langle y|x\right\rangle :=\left\langle x,y\right\rangle }$ is used in physics). For every continuous linear functional ${\displaystyle \varphi \in H^{*},}$ there exists a unique ${\displaystyle f_{\varphi }\in H}$ such that

${\displaystyle \varphi (x)=\left\langle f_{\varphi }|x\right\rangle =\left\langle x,f_{\varphi }\right\rangle }$      for all ${\displaystyle x\in H,}$

and moreover,

${\displaystyle \left\|f_{\varphi }\right\|_{H}=\|\varphi \|_{H^{*}}.}$
• Importantly for complex Hilbert spaces, note that the vector ${\displaystyle f_{\varphi }\in H}$ is always located in the antilinear coordinate of the inner product (no matter which notation is used).[note 1]

Consequently, the map ${\displaystyle H^{*}\to H}$ defined by ${\displaystyle \varphi \mapsto f_{\varphi }}$ is a bijective antilinear isometry whose inverse is the antilinear isometry ${\displaystyle \Phi :H\to H^{*}}$ defined by ${\displaystyle y\mapsto \left\langle \bullet ,y\right\rangle =\left\langle y|\bullet \right\rangle .}$ For ${\displaystyle y\in H,}$ the physics notation for the functional ${\displaystyle \Phi (y)\in H^{*}}$ is the bra ${\displaystyle \left\langle y\right|,}$ where explicitly this means that ${\displaystyle \left\langle y\right|:=\Phi (y),}$ which complements the ket notation ${\displaystyle \left|y\right\rangle }$ defined by ${\displaystyle \left|y\right\rangle :=y.}$

Proof —

Let ${\displaystyle M:=\operatorname {ker} \varphi :=\{u\in H\ |\ \varphi (u)=0\}.}$ Then ${\displaystyle M}$ is closed subspace of ${\displaystyle H.}$ If ${\displaystyle M=H}$ (or equivalently, if φ = 0) then we take ${\displaystyle f_{\varphi }:=0}$ and we're done. So assume ${\displaystyle M\neq H.}$

It is first shown that ${\displaystyle M^{\perp }=\{v\in H~:~\langle m,v\rangle =0~{\text{ for all }}m\in M\}}$ is one-dimensional. Using Zorn's lemma or the well-ordering theorem it can be shown that there exists some non-zero vector ${\displaystyle v}$ in ${\displaystyle M^{\perp }}$ — proving this is left as an exercise to the reader. We continue: Let ${\displaystyle v_{1},}$ and ${\displaystyle v_{2}}$ be nonzero vectors in ${\displaystyle M^{\perp }.}$ Then ${\displaystyle \varphi (v_{1})\neq 0,}$ and ${\displaystyle \varphi (v_{2})\neq 0,}$ and there must exist a nonzero real number ${\displaystyle \lambda \neq 0}$ such that ${\displaystyle \lambda \varphi (v_{1})=\varphi (v_{2}).}$ This implies that ${\displaystyle \lambda v_{1}-v_{2}\in M^{\perp }}$ and ${\displaystyle \varphi (\lambda v_{1}-v_{2})=0,}$ so ${\displaystyle \lambda v_{1}-v_{2}\in M.}$ Since ${\displaystyle M^{\perp }\cap M=\{0\},}$ this implies that ${\displaystyle \lambda v_{1}-v_{2}=0,}$ as desired.

Now let ${\displaystyle g}$ be a unit vector in ${\displaystyle M^{\perp }.}$ For arbitrary ${\displaystyle x\in H,}$ let ${\displaystyle v}$ be the orthogonal projection of ${\displaystyle x}$ onto ${\displaystyle M^{\perp }.}$ Then ${\displaystyle v=\langle x,g\rangle g}$ and ${\displaystyle \langle g,x-v\rangle =0}$ (from the properties of orthogonal projections), so that ${\displaystyle x-v\in M}$ and ${\displaystyle \langle x,g\rangle =\langle v,g\rangle .}$ Thus

${\displaystyle \varphi (x)=\varphi (v+x-v)=\varphi (\langle x,g\rangle g)+\varphi (x-v)=\langle x,g\rangle \varphi (g)+0=\langle x,g\rangle \varphi (g)=\left\langle x,{\overline {\varphi (g)}}g\right\rangle .}$

Because of this, we take ${\displaystyle f_{\varphi }:={\overline {\varphi (g)}}g.}$ We also see that ${\displaystyle \left\|f_{\varphi }\right\|_{H}=|\varphi (g)|.}$

From the Cauchy-Bunyakovsky-Schwarz inequality ${\displaystyle |\varphi (x)|\leq \|x\||\varphi (g)|\|g\|=\|x\||\varphi (g)|,}$ and so if ${\displaystyle x}$ has unit norm then ${\displaystyle \|\varphi \|_{H^{*}}\leq |\varphi (g)|.}$ Since ${\displaystyle g}$ has unit norm, we have ${\displaystyle \|\varphi \|_{H*}=|\varphi (g)|.}$

Observations:

• ${\displaystyle \varphi \left(f_{\varphi }\right)=\left\langle f_{\varphi },f_{\varphi }\right\rangle =\left\|f_{\varphi }\right\|^{2}=\left\|\varphi \right\|^{2}.}$ So in particular, we always have ${\displaystyle \varphi \left(f_{\varphi }\right)\geq 0}$ is real, where ${\displaystyle \varphi \left(f_{\varphi }\right)=0}$ fφ = 0 φ = 0.
• Showing that there is a non-zero vector ${\displaystyle v}$ in ${\displaystyle M^{\perp }}$ relies on the continuity of ${\displaystyle \phi }$ and the Cauchy completeness of ${\displaystyle H}$. This is the only place in the proof in which these properties are used.

Constructions

Using the notation from the theorem above, we now provide ways of constructing ${\displaystyle f_{\varphi }}$ from ${\displaystyle \varphi \in H^{*}.}$

• If φ = 0 then fφ := 0 and otherwise ${\displaystyle f_{\varphi }:={\frac {{\overline {\varphi (g)}}g}{\|g\|^{2}}}}$ for any ${\displaystyle 0\neq g\in \left(\operatorname {ker} \varphi \right)^{\perp }.}$
• If ${\displaystyle g\in \left(\operatorname {ker} \varphi \right)^{\perp }}$ is a unit vector then ${\displaystyle f_{\varphi }:={\overline {\varphi (g)}}g.}$
• If g is a unit vector satisfying the above condition then the same is true of -g (the only other unit vector in ${\displaystyle \left(\operatorname {ker} \varphi \right)^{\perp }}$). However, ${\displaystyle {\overline {\varphi (-g)}}(-g)={\overline {\varphi (g)}}g=f_{\varphi }}$ so both these vectors result in the same ${\displaystyle f_{\varphi }.}$
• If φ(x) ≠ 0 and xK is the orthogonal projection of ${\displaystyle x}$ onto ker φ, then ${\displaystyle f_{\varphi }={\frac {\|\varphi \|^{2}}{\varphi (x)}}(x-x_{K}).}$[note 2]
• Suppose φ ≠ 0 and let ${\displaystyle M_{\mathbb {R} }:=(\operatorname {Re} \varphi )^{-1}\left(0\right)=\varphi ^{-1}\left(i\mathbb {R} \right)}$ where note that ${\displaystyle f_{\varphi }\not \in M_{\mathbb {R} }}$ since ${\displaystyle \varphi \left(f_{\varphi }\right)=\left\|\varphi \right\|^{2}\neq 0}$ is real and ${\displaystyle \operatorname {ker} \varphi }$ is a proper subset of ${\displaystyle M_{\mathbb {R} }.}$ If we reinterpret ${\displaystyle H}$ as a real Hilbert space H (with the usual real-valued inner product defined by ${\displaystyle \left\langle x,y\right\rangle _{\mathbb {R} }:=\operatorname {Re} \left\langle x,y\right\rangle }$), then ${\displaystyle \operatorname {ker} \varphi }$ has real codimension 1 in ${\displaystyle M_{\mathbb {R} },}$ where ${\displaystyle M_{\mathbb {R} }}$ has real codimension 1 in H, and ${\displaystyle \left\langle f_{\varphi },M_{\mathbb {R} }\right\rangle _{\mathbb {R} }=0}$ (i.e. ${\displaystyle f_{\varphi }}$ is perpendicular to ${\displaystyle M_{\mathbb {R} }}$ with respect to ${\displaystyle \left\langle \cdot ,\cdot \right\rangle _{\mathbb {R} }}$).
• In the theorem and constructions above, if we replace ${\displaystyle H}$ with its real Hilbert space counterpart H and if we replace φ with Re φ then ${\displaystyle f_{\varphi }=f_{\operatorname {Re} \varphi },}$ meaning that we will obtain the exact same vector ${\displaystyle f_{\varphi }}$ 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 ${\displaystyle \varphi \in H^{*},}$ the corresponding element ${\displaystyle f_{\varphi }\in H}$ can be constructed uniquely by
${\displaystyle f_{\varphi }=\varphi (e_{1})e_{1}+\varphi (e_{2})e_{2}+...,}$
where ${\displaystyle \{e_{i}\}}$ is an orthonormal basis of H, and the value of ${\displaystyle f_{\varphi }}$ does not vary by choice of basis. Thus, if ${\displaystyle y\in H,y=a_{1}e_{1}+a_{2}e_{2}+...,}$ then
${\displaystyle \varphi (y)=a_{1}\varphi (e_{1})+a_{2}\varphi (e_{2})+...=\langle f_{\varphi },y\rangle .}$

Canonical injection from a Hilbert space to its dual and anti-dual

For every ${\displaystyle y\in H,}$ the inner product on ${\displaystyle H}$ can be used to define two continuous (i.e. bounded) canonical maps:

• The map defined by placing ${\displaystyle y}$ into the antilinear coordinate of the inner product and letting the variable ${\displaystyle h\in H}$ 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 ${\displaystyle H^{*},}$ which is the continuous dual space of ${\displaystyle H.}$ The canonical map from ${\displaystyle H}$ into its dual ${\displaystyle H^{*}}$[1] is the antilinear operator
${\displaystyle \Phi :=\operatorname {In} _{H}^{H^{*}}~:~H\to H^{*}}$       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 ${\displaystyle H}$ can be written (uniquely) in this form.[1]
• The map defined by placing ${\displaystyle y}$ into the linear coordinate of the inner product and letting the variable ${\displaystyle h\in H}$ 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 ${\displaystyle {\overline {H}}^{*},}$ which is the continuous anti-dual space of ${\displaystyle H.}$ The canonical map from ${\displaystyle H}$ into its anti-dual ${\displaystyle {\overline {H}}^{*}}$[1] is the linear operator
${\displaystyle \operatorname {In} _{H}^{{\overline {H}}^{*}}~:~H\to {\overline {H}}^{*}}$       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 ${\displaystyle H}$ can be written (uniquely) in this form.[1]

If ${\displaystyle \operatorname {Cong} :H^{*}\to {\overline {H}}^{*}}$ is the canonical antilinear bijective isometry ${\displaystyle f\mapsto {\overline {f}}}$ that was defined above, then the following equality holds:

${\displaystyle \operatorname {Cong} ~\circ ~\operatorname {In} _{H}^{H^{*}}~=~\operatorname {In} _{H}^{{\overline {H}}^{*}}.}$

Let ${\displaystyle A:H\to Z}$ be a continuous linear operator between Hilbert spaces ${\displaystyle \left(H,\langle \cdot ,\cdot \rangle _{H}\right)}$ and ${\displaystyle \left(Z,\langle \cdot ,\cdot \rangle _{Z}\right).}$ As before, let ${\displaystyle \langle y|x\rangle _{H}:=\langle x,y\rangle _{H}}$ and ${\displaystyle \langle y|x\rangle _{Z}:=\langle x,y\rangle _{Z}.}$ The adjoint of ${\displaystyle A:H\to Z}$ is the linear operator ${\displaystyle A^{*}:Z\to H}$ defined by the condition:

${\displaystyle \left\langle z|Ah\right\rangle _{Z}=\left\langle A^{*}z|h\right\rangle _{H},}$      for all ${\displaystyle h\in H}$ and all ${\displaystyle z\in Z.}$

It is also possible to define the transpose of ${\displaystyle A:H\to Z,}$ which is the map ${\displaystyle {}^{t}A:Z^{*}\to H^{*}}$ defined by sending a continuous linear functionals ${\displaystyle \psi \in Z^{*}}$ to

${\displaystyle {}^{t}A(\psi ):=\psi \circ A}$

The adjoint ${\displaystyle A^{*}:Z\to H}$ is actually just to the transpose ${\displaystyle {}^{t}A:Z^{*}\to H^{*}}$ when the Riesz representation theorem is used to identify ${\displaystyle Z}$ with ${\displaystyle Z^{*}}$ and ${\displaystyle H}$ with ${\displaystyle H^{*}.}$ To make this explicit, let ${\displaystyle \Phi _{H}~:~H\to H^{*}}$ and ${\displaystyle \Phi _{Z}~:~Z\to Z^{*}}$ be the bijective antilinear isometries defined respectively by

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

so that by definition

${\displaystyle (\Phi _{H}g)h=\langle g|h\rangle _{H}=\langle h,g\rangle _{H}}$ for all ${\displaystyle g,h\in H}$      and      ${\displaystyle (\Phi _{Z}z)y=\langle z|y\rangle _{Z}=\langle y,z\rangle _{Z}}$ for all ${\displaystyle y,z\in Z.}$

The relationship between the adjoint and transpose can be shown (see footnote for proof)[note 3] to be:

${\displaystyle {}^{t}A~\circ ~\Phi _{Z}~=~\Phi _{H}~\circ ~A^{*}}$

which can be rewritten as:

${\displaystyle A^{*}~=~\Phi _{H}^{-1}~\circ ~{}^{t}A~\circ ~\Phi _{Z}}$      and      ${\displaystyle {}^{t}A~=~\Phi _{H}~\circ ~A^{*}~\circ ~\Phi _{Z}^{-1}.}$

Extending the bra-ket notation to bras and kets

Let ${\displaystyle \left(H,\langle \cdot ,\cdot \rangle _{H}\right)}$ be a Hilbert space and as before, let ${\displaystyle \langle y|x\rangle _{H}:=\langle x,y\rangle _{H}.}$ Let ${\displaystyle \Phi ~:~H\to H^{*}}$ be the bijective antilinear isometry defined by

gg | • ⟩H = ⟨ •, gH

so that by definition

${\displaystyle (\Phi h)g=\langle h|g\rangle _{H}=\langle g,h\rangle _{H}}$      for all ${\displaystyle g,h\in H.}$
Bras

Given a vector ${\displaystyle h\in H,}$ let ${\displaystyle \langle h|}$ denote the continuous linear functional ${\displaystyle \Phi h}$; that is, ${\displaystyle \langle h|~:=~\Phi h.}$ The resulting of plugging some given ${\displaystyle g\in H}$ into the functional ${\displaystyle \langle h|}$ is the scalar ${\displaystyle \langle h|g\rangle _{H}=\langle g,h\rangle _{H},}$ where ${\displaystyle \langle h|g\rangle }$ is the notation that is used instead of ${\displaystyle \langle h|(g)}$ or ${\displaystyle \langle h|g.}$ The assignment ${\displaystyle h\mapsto \langle h|}$ is just the isometric antilinear isomorphism ${\displaystyle \Phi ~:~H\to H^{*}}$ so ${\displaystyle ~\langle cg+h|~=~{\overline {c}}\langle g|~+~\langle h|~}$ holds for all ${\displaystyle g,h\in H}$ and all scalars ${\displaystyle c.}$

Given a continuous linear functional ${\displaystyle \psi \in H^{*},}$ let ${\displaystyle \langle \psi |}$ denote the vector ${\displaystyle \Phi ^{-1}\psi }$; that is, ${\displaystyle \langle \psi |~:=~\Phi ^{-1}\psi .}$ The defining condition of the vector ${\displaystyle \langle \psi |\in H}$ is the technically correct but unsightly equality

${\displaystyle \left\langle \,\langle \psi |\,|g\right\rangle _{H}~=~\psi g}$      for all ${\displaystyle g\in H,}$

which is why the notation ${\displaystyle \left\langle \psi |g\right\rangle }$ is used in place of ${\displaystyle \left\langle \,\langle \psi |\,|g\right\rangle _{H}=\left\langle g,\,\langle \psi |\,\right\rangle _{H}.}$ The defining condition becomes

${\displaystyle \left\langle \psi |g\right\rangle ~=~\psi g}$      for all ${\displaystyle g\in H.}$

The assignment ${\displaystyle \psi \mapsto \langle \psi |}$ is just the isometric antilinear isomorphism ${\displaystyle \Phi ^{-1}~:~H^{*}\to H}$ so ${\displaystyle ~\langle c\psi +\phi |~=~{\overline {c}}\langle \psi |~+~\langle \phi |~}$ holds for all ${\displaystyle \phi ,\psi \in H^{*}}$ and all scalars ${\displaystyle c.}$

Kets

For any given vector ${\displaystyle g\in H,}$ the notation ${\displaystyle |g\rangle }$ is used to denote ${\displaystyle g}$; that is, ${\displaystyle |g\rangle :=g.}$ The notation ${\displaystyle \langle h|g\rangle }$ and ${\displaystyle \langle \psi |g\rangle }$ is used in place of ${\displaystyle \left\langle h|\,|g\rangle \,\right\rangle _{H}~=~\left\langle \,|g\rangle ,h\right\rangle _{H}}$ and ${\displaystyle \left\langle \psi |\,|g\rangle \,\right\rangle _{H}~=~\left\langle g,\,\langle \psi |\,\right\rangle _{H},}$ respectively. As expected, ${\displaystyle ~\langle \psi |g\rangle ~=~\psi g~}$ and ${\displaystyle ~\langle h|g\rangle ~}$ really is just the scalar ${\displaystyle ~\langle h|g\rangle _{H}~=~\langle g,h\rangle _{H}.}$

Properties of induced antilinear map

The mapping ${\displaystyle \Phi }$: HH* defined by ${\displaystyle \Phi (x)}$ = ${\displaystyle \varphi _{x}}$ is an isometric antilinear isomorphism, meaning that:

• ${\displaystyle \Phi }$ is bijective.
• The norms of ${\displaystyle x}$ and ${\displaystyle \varphi _{x}}$ agree: ${\displaystyle \Vert x\Vert =\Vert \Phi (x)\Vert .}$
• Using this fact, this map could be used to give an equivalent definition of the canonical dual norm of ${\displaystyle H^{*}.}$ The canonical inner product on ${\displaystyle H^{*}}$ could be defined similarly.
• ${\displaystyle \Phi }$ is additive: ${\displaystyle \Phi (x_{1}+x_{2})=\Phi (x_{1})+\Phi (x_{2}).}$
• If the base field is ${\displaystyle \mathbb {R} ,}$ then ${\displaystyle \Phi (\lambda x)=\lambda \Phi (x)}$ for all real numbers λ.
• If the base field is ${\displaystyle \mathbb {C} ,}$ then ${\displaystyle \Phi (\lambda x)={\bar {\lambda }}\Phi (x)}$ for all complex numbers λ, where ${\displaystyle {\bar {\lambda }}}$ denotes the complex conjugation of ${\displaystyle \lambda .}$

The inverse map of ${\displaystyle \Phi }$ can be described as follows. Given a non-zero element ${\displaystyle \varphi }$ of H*, the orthogonal complement of the kernel of ${\displaystyle \varphi }$ is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set ${\displaystyle x={\overline {\varphi (z)}}\cdot z/{\left\Vert z\right\Vert }^{2}.}$ Then ${\displaystyle \Phi (x)}$ = ${\displaystyle \varphi .}$

Alternatively, the assignment ${\displaystyle x\mapsto \varphi _{x}}$ can be viewed as a bijective linear isometry ${\displaystyle H\to {\overline {H}}^{*}}$ into the anti-dual space of ${\displaystyle H,}$[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 ${\displaystyle \langle \psi |}$ has a corresponding ket ${\displaystyle |\psi \rangle ,}$ and the latter is unique.

Notes

1. Trèves 2006, pp. 112-123.
1. ^ If ${\displaystyle \mathbb {F} =\mathbb {R} }$ then the inner product will be symmetric so it doesn't matter which coordinate of the inner product the element ${\displaystyle y}$ is placed into because the same map will result. But if ${\displaystyle \mathbb {F} =\mathbb {C} }$ then except for the constant 0 map, antilinear functionals on ${\displaystyle H}$ are completely distinct from linear functionals on ${\displaystyle H,}$ which makes the coordinate that ${\displaystyle y}$ is placed into is very important. For a non-zero ${\displaystyle y\in H}$ to induce a linear functional (rather than an antilinear functional), ${\displaystyle y}$ 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 ${\displaystyle h\mapsto \left\langle y,h\right\rangle =\langle h|y\rangle ,}$ which is not a linear functional on ${\displaystyle H}$ and so it will not be an element of the continuous dual space ${\displaystyle H^{*}.}$
2. ^ Since we must have ${\displaystyle x_{K}=x-{\frac {\left\langle x,f_{\varphi }\right\rangle }{\left\|f_{\varphi }\right\|^{2}}}f_{\varphi }.}$ Now use ${\displaystyle \left\|f_{\varphi }\right\|^{2}=\left\|\varphi \right\|^{2}}$ and ${\displaystyle \left\langle x,f_{\varphi }\right\rangle =\varphi \left(x\right)}$ and solve for ${\displaystyle f_{\varphi }.}$
3. ^ To show that ${\displaystyle {}^{t}A~\circ ~\Phi _{Z}~=~\Phi _{H}~\circ ~A^{*},}$ fix ${\displaystyle z\in Z.}$ The definition of ${\displaystyle {}^{t}A}$ implies ${\displaystyle \left({}^{t}A\circ \Phi _{Z}\right)z=\left({}^{t}A(\Phi _{Z}z)\right)=\left(\Phi _{Z}z\right)\circ A}$ so it remains to show that ${\displaystyle \left(\Phi _{Z}z\right)\circ A=\Phi _{H}\left(A^{*}z\right).}$ If ${\displaystyle h\in H}$ then ${\displaystyle \left(\left(\Phi _{Z}z\right)\circ A\right)h=\left(\Phi _{Z}z\right)(Ah)=\langle z|Ah\rangle _{Z}=\langle A^{*}z|h\rangle _{H}=\left(\Phi _{H}\left(A^{*}z\right)\right)h,}$ as desired. ◼

References

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