Editing Riesz representation theorem (section)

=== Canonical injections into the dual and anti-dual ===

'''Induced linear map into anti-dual'''

The map defined by placing <math>y</math> into the {{em|linear}} coordinate of the inner product and letting the variable <math>h \in H</math> vary over the {{em|antilinear}} coordinate results in an [[Antilinear map|{{em|antilinear}} functional]]:
<math display=block>\langle \,\cdot \mid y\, \rangle = \langle \,y, \cdot\, \rangle : H \to \mathbb{F} \quad \text{ defined by } \quad h \mapsto \langle \,h \mid y\, \rangle = \langle \,y, h\, \rangle.</math>

This map is an element of <math>\overline{H}^*,</math> which is the continuous [[anti-dual space]] of <math>H.</math> 
The {{em|canonical map from <math>H</math> into its anti-dual}} <math>\overline{H}^*</math>{{sfn|Trèves|2006|pp=112–123}} is the [[Linear operator|{{em|linear}} operator]] 
<math display=block>\begin{alignat}{4}
\operatorname{In}_H^{\overline{H}^*} :\;&& H &&\;\to    \;& \overline{H}^* \\[0.3ex]
                                        && y &&\;\mapsto\;& \langle \,\cdot \mid y\, \rangle = \langle \,y, \cdot\, \rangle \\[0.3ex]
\end{alignat}</math>
which is also an [[Injective map|injective]] [[isometry]].{{sfn|Trèves|2006|pp=112–123}} 
The [[Fundamental theorem of Hilbert spaces]], which is related to Riesz representation theorem, states that this map is surjective (and thus [[Bijective map|bijective]]). Consequently, every antilinear functional on <math>H</math> can be written (uniquely) in this form.{{sfn|Trèves|2006|pp=112–123}} 

If <math>\operatorname{Cong} : H^* \to \overline{H}^*</math> is the canonical [[Antilinear map|{{em|anti}}linear]] [[Bijective map|bijective]] [[isometry]] <math>f \mapsto \overline{f}</math> that was defined above, then the following equality holds:
<math display=block>\operatorname{Cong} ~\circ~ \operatorname{In}_H^{H^*} ~=~ \operatorname{In}_H^{\overline{H}^*}.</math>