Editing Riesz representation theorem (section)

== Preliminaries and notation ==

Let <math>H</math> be a [[Hilbert space]] over a field <math>\mathbb{F},</math> where <math>\mathbb{F}</math> is either the real numbers <math>\R</math> or the complex numbers <math>\Complex.</math> If <math>\mathbb{F} = \Complex</math> (resp. if <math>\mathbb{F} = \R</math>) then <math>H</math> is called a {{em|complex Hilbert space}} (resp. a {{em|real Hilbert space}}). Every real Hilbert space can be extended to be a [[dense subset]] of a unique (up to [[Bijective map|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 [[Mathematician|mathematicians]] and [[Physicist|physicists]] and will describe the theorem for both. 
In both mathematics and physics, if a Hilbert space is assumed to be real (that is, if <math>\mathbb{F} = \R</math>) 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 {{em|or}} complex Hilbert space. 

=== Linear and antilinear maps ===

By definition, an [[Antilinear map|{{em|antilinear map}}]] (also called a {{em|conjugate-linear map}}) <math>f : H \to Y</math> is a map between [[vector space]]s that is {{em|[[Additive map|additive]]}}:
<math display=block>f(x + y) = f(x) + f(y) \quad \text{ for all } x, y \in H,</math>
and {{em|antilinear}} (also called {{em|conjugate-linear}} or {{em|conjugate-homogeneous}}):
<math display="block">f(c x) = \overline{c} f(x) \quad \text{ for all } x \in H \text{ and all scalar } c \in \mathbb{F},</math>
where <math>\overline{c}</math> is the conjugate of the complex number <math>c = a + b i</math>, given by <math>\overline{c} = a - b i</math>.

In contrast, a map <math>f : H \to Y</math> is [[Linear map|linear]] if it is additive and [[Homogeneous function|{{em|homogeneous}}]]: 
<math display=block>f(c x) = c f(x) \quad \text{ for all } x \in H \quad \text{ and all scalars } c \in \mathbb{F}.</math>

Every constant <math>0</math> map is always both linear and antilinear. If <math>\mathbb{F} = \R</math> 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 linear operator|continuous]] if and only if it is [[Bounded linear operator|bounded]]; the same is true of antilinear maps. The [[Inverse function|inverse]] of any antilinear (resp. linear) bijection is again an antilinear (resp. linear) bijection. The composition of two {{em|anti}}linear maps is a {{em|linear}} map. 

'''Continuous dual and anti-dual spaces'''

A {{em|functional}} on <math>H</math> is a function <math>H \to \mathbb{F}</math> whose [[codomain]] is the underlying scalar field <math>\mathbb{F}.</math> 
Denote by <math>H^*</math> (resp. by <math>\overline{H}^*)</math> the set of all continuous linear (resp. continuous antilinear) functionals on <math>H,</math> which is called the {{em|[[Continuous dual space|(continuous) dual space]]}} (resp. the {{em|[[Anti-dual space|(continuous) anti-dual space]]}}) of <math>H.</math>{{sfn|Trèves|2006|pp=112–123}} 
If <math>\mathbb{F} = \R</math> then linear functionals on <math>H</math> are the same as antilinear functionals and consequently, the same is true for such continuous maps: that is, <math>H^* = \overline{H}^*.</math> 

'''One-to-one correspondence between linear and antilinear functionals'''

Given any functional <math>f ~:~ H \to \mathbb{F},</math> the {{em|conjugate of <math>f</math>}} is the functional 
<math display=block>\begin{alignat}{4}
\overline{f} : \,& H && \to    \,&& \mathbb{F} \\
                 & h && \mapsto\,&& \overline{f(h)}. \\
\end{alignat}</math>

This assignment is most useful when <math>\mathbb{F} = \Complex</math> because if <math>\mathbb{F} = \R</math> then <math>f = \overline{f}</math> and the assignment <math>f \mapsto \overline{f}</math> reduces down to the [[identity map]]. 

The assignment <math>f \mapsto \overline{f}</math> defines an antilinear [[Bijective map|bijective]] correspondence from the set of 
:all functionals (resp. all linear functionals, all continuous linear functionals <math>H^*</math>) on <math>H,</math> 
onto the set of 
:all functionals (resp. all {{em|anti}}linear functionals, all continuous {{em|anti}}linear functionals <math>\overline{H}^*</math>) on <math>H.</math>

=== Mathematics vs. physics notations and definitions of inner product ===

The [[Hilbert space]] <math>H</math> has an associated [[inner product]] <math>H \times H \to \mathbb{F}</math> valued in <math>H</math>'s underlying scalar field <math>\mathbb{F}</math> that is linear in one coordinate and antilinear in the other (as specified below).
If <math>H</math> is a complex Hilbert space (<math>\mathbb{F} = \Complex</math>), then there is a crucial difference between the notations prevailing in mathematics versus physics, regarding which of the two variables is linear.
However, for real Hilbert spaces (<math>\mathbb{F} = \R</math>), the inner product is a [[Symmetric map|symmetric]] map that is linear in each coordinate ([[bilinear map|bilinear]]), so there can be no such confusion. 

In [[mathematics]], the inner product on a Hilbert space <math>H</math> is often denoted by <math>\left\langle \cdot\,, \cdot \right\rangle</math> or <math>\left\langle \cdot\,, \cdot \right\rangle_H</math> while in [[physics]], the [[bra–ket notation]] <math>\left\langle \cdot \mid \cdot \right\rangle</math> or <math>\left\langle \cdot \mid \cdot \right\rangle_H</math> is typically used. In this article, these two notations will be related by the equality:
<math display="block">\left\langle x, y \right\rangle := \left\langle y \mid x \right\rangle \quad \text{ for all } x, y \in H.</math>These have the following properties:<ol>
<li>The map <math>\left\langle \cdot\,, \cdot \right\rangle</math> is ''linear in its first coordinate''; equivalently, the map <math>\left\langle \cdot \mid \cdot \right\rangle</math> is ''linear in its second coordinate''. That is, for fixed <math>y \in H,</math> the map 
<math>\left\langle \,y\mid \cdot\, \right\rangle = \left\langle \,\cdot\,, y\, \right\rangle : H \to \mathbb{F}</math> with <math display="inline">h \mapsto \left\langle \,y\mid h\, \right\rangle = \left\langle \,h, y\, \right\rangle </math>
is a linear functional on <math>H.</math> This linear functional is continuous, so <math>\left\langle \,y\mid\cdot\, \right\rangle = \left\langle \,\cdot, y\, \right\rangle \in H^*.</math>
</li>
<li>The map <math>\left\langle \cdot\,, \cdot \right\rangle</math> is ''[[Antilinear map|antilinear]] in its {{em|second}} coordinate''; equivalently, the map <math>\left\langle \cdot \mid \cdot \right\rangle</math> is ''antilinear in its {{em|first}} coordinate''. That is, for fixed <math>y \in H,</math> the map
<math>\left\langle \,\cdot\mid y\, \right\rangle = \left\langle \,y, \cdot\, \right\rangle : H \to \mathbb{F}</math> with <math display="inline">h \mapsto \left\langle \,h\mid y\, \right\rangle = \left\langle \,y, h\, \right\rangle </math>
is an antilinear functional on <math>H.</math> This antilinear functional is continuous, so <math>\left\langle \,\cdot\mid y\, \right\rangle = \left\langle \,y, \cdot\, \right\rangle \in \overline{H}^*.</math>
</li>
</ol>

In computations, one must consistently use either the mathematics notation <math>\left\langle \cdot\,, \cdot \right\rangle</math>, which is (linear, antilinear); or the physics notation <math>\left\langle \cdot \mid \cdot \right\rangle</math>, which is (antilinear | linear).

=== Canonical norm and inner product on the dual space and anti-dual space ===

If <math>x = y</math> then <math>\langle \,x\mid x\, \rangle = \langle \,x, x\, \rangle</math> is a non-negative real number and the map
<math display=block>\|x\| := \sqrt{\langle x, x \rangle} = \sqrt{\langle x \mid x \rangle}</math> 

defines a [[Norm (mathematics)|canonical norm]] on <math>H</math> that makes <math>H</math> into a [[normed space]].{{sfn|Trèves|2006|pp=112–123}} 
As with all normed spaces, the (continuous) dual space <math>H^*</math> carries a canonical norm, called the {{em|[[dual norm]]}}, that is defined by{{sfn|Trèves|2006|pp=112–123}}
<math display=block>\|f\|_{H^*} ~:=~ \sup_{\|x\| \leq 1, x \in H} |f(x)| \quad \text{ for every } f \in H^*.</math> 

The canonical norm on the (continuous) [[anti-dual space]] <math>\overline{H}^*,</math> denoted by <math>\|f\|_{\overline{H}^*},</math> is defined by using this same equation:{{sfn|Trèves|2006|pp=112–123}} 
<math display=block>\|f\|_{\overline{H}^*} ~:=~ \sup_{\|x\| \leq 1, x \in H} |f(x)| \quad \text{ for every } f \in \overline{H}^*.</math> 

This canonical norm on <math>H^*</math> satisfies the [[parallelogram law]], which means that the [[polarization identity]] can be used to define a {{em|canonical inner product on <math>H^*,</math>}} which this article will denote by the notations 
<math display=block>\left\langle f, g \right\rangle_{H^*} := \left\langle g \mid f \right\rangle_{H^*},</math> 
where this inner product turns <math>H^*</math> into a Hilbert space. There are now two ways of defining a norm on <math>H^*:</math> the norm induced by this inner product (that is, the norm defined by <math>f \mapsto \sqrt{\left\langle f, f \right\rangle_{H^*}}</math>) and the usual [[dual norm]] (defined as the supremum over the closed unit ball). These norms are the same; explicitly, this means that the following holds for every <math>f \in H^*:</math>
<math display=block>\sup_{\|x\| \leq 1, x \in H} |f(x)| = \|f\|_{H^*} ~=~ \sqrt{\langle f, f \rangle_{H^*}} ~=~ \sqrt{\langle f \mid f \rangle_{H^*}}.</math> 

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 <math>H^*.</math> 

The same equations that were used above can also be used to define a norm and inner product on <math>H</math>'s [[anti-dual space]] <math>\overline{H}^*.</math>{{sfn|Trèves|2006|pp=112–123}} 

'''Canonical isometry between the dual and antidual'''

The [[complex conjugate]] <math>\overline{f}</math> of a functional <math>f,</math> which was defined above, satisfies 
<math display=block>\|f\|_{H^*} ~=~ \left\|\overline{f}\right\|_{\overline{H}^*} \quad \text{ and } \quad \left\|\overline{g}\right\|_{H^*} ~=~ \|g\|_{\overline{H}^*}</math>
for every <math>f \in H^*</math> and every <math>g \in \overline{H}^*.</math> 
This says exactly that the canonical antilinear [[Bijective map|bijection]] defined by
<math display=block>\begin{alignat}{4}
\operatorname{Cong} :\;&& H^* &&\;\to    \;& \overline{H}^* \\[0.3ex]
                       && f   &&\;\mapsto\;& \overline{f} \\
\end{alignat}</math>
as well as its inverse <math>\operatorname{Cong}^{-1} ~:~ \overline{H}^* \to H^*</math> are antilinear [[Isometry|isometries]] and consequently also [[homeomorphism]]s. 
The inner products on the dual space <math>H^*</math> and the anti-dual space <math>\overline{H}^*,</math> denoted respectively by <math>\langle \,\cdot\,, \,\cdot\, \rangle_{H^*}</math> and <math>\langle \,\cdot\,, \,\cdot\, \rangle_{\overline{H}^*},</math> are related by
<math display=block>\langle \,\overline{f}\, | \,\overline{g}\, \rangle_{\overline{H}^*} = \overline{\langle \,f\, | \,g\, \rangle_{H^*}} = \langle \,g\, | \,f\, \rangle_{H^*} \qquad \text{ for all } f, g \in H^*</math>
and
<math display=block>\langle \,\overline{f}\, | \,\overline{g}\, \rangle_{H^*} = \overline{\langle \,f\, | \,g\, \rangle_{\overline{H}^*}} = \langle \,g\, | \,f\, \rangle_{\overline{H}^*} \qquad \text{ for all } f, g \in \overline{H}^*.</math>

If <math>\mathbb{F} = \R</math> then <math>H^* = \overline{H}^*</math> and this canonical map <math>\operatorname{Cong} : H^* \to \overline{H}^*</math> reduces down to the identity map.