Editing Riesz representation theorem (section)

=== 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.