Riesz representation theorem

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 ${\displaystyle H.}$

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.

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.

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 f_φ := 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 x_K 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_ℝ (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_ℝ, 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_ℝ 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_ℝ, ⟨⋅, ⋅⟩_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 ${\displaystyle \varphi \in H^{*},}$ 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

  ${\displaystyle \varphi (y)=a_{1}\varphi (e_{1})+a_{2}\varphi (e_{2})+...=\langle f_{\varphi },y\rangle .}$