Editing Riesz representation theorem (section)

=== 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>