Editing Antilinear map (section)

== Definitions and characterizations ==

A function is called {{em|antilinear}} or {{em|conjugate linear}} if it is [[Additive map|additive]] and [[conjugate homogeneous]]. An {{em|antilinear functional}} on a vector space <math>V</math> is a scalar-valued antilinear map. 

A function <math>f</math> is called {{em|[[Additive map|additive]]}} if
<math display=block>f(x + y) = f(x) + f(y) \quad \text{ for all vectors } x, y</math>
while it is called {{em|[[Conjugate homogeneity|conjugate homogeneous]]}} if 
<math display=block>f(ax) = \overline{a} f(x) \quad \text{ for all vectors } x \text{ and all scalars } a.</math> 
In contrast, a linear map is a function that is additive and [[homogeneous]], where <math>f</math> is called {{em|homogeneous}} if
<math display=block>f(ax) = a f(x) \quad \text{ for all vectors } x \text{ and all scalars } a.</math> 

An antilinear map <math>f : V \to W</math> may be equivalently described in terms of the [[linear map]] <math>\overline{f} : V \to \overline{W}</math> from <math>V</math> to the [[complex conjugate vector space]] <math>\overline{W}.</math>

=== Examples ===

==== Anti-linear dual map ====

Given a complex vector space <math>V</math> of rank 1, we can construct an anti-linear dual map which is an anti-linear map <math display="block">l:V \to \Complex</math> sending an element <math>x_1 + iy_1</math> for <math>x_1,y_1 \in \R</math> to <math display="block">x_1 + iy_1 \mapsto a_1 x_1 - i b_1 y_1</math> for some fixed real numbers <math>a_1,b_1.</math> We can extend this to any finite dimensional complex vector space, where if we write out the standard basis <math>e_1, \ldots, e_n</math> and each standard basis element as <math display="block">e_k = x_k + iy_k</math> then an anti-linear complex map to <math>\Complex</math> will be of the form <math display="block">\sum_k x_k + iy_k \mapsto \sum_k a_k x_k - i b_k y_k</math> for <math>a_k,b_k \in \R.</math>

==== Isomorphism of anti-linear dual with real dual ====

The anti-linear dual<ref name=":0">{{Cite book|last=Birkenhake|first=Christina| url=https://www.worldcat.org/oclc/851380558 | title=Complex Abelian Varieties | date=2004 | publisher=Springer Berlin Heidelberg|others=Herbert Lange |isbn=978-3-662-06307-1| edition=Second, augmented| location=Berlin, Heidelberg| oclc=851380558}}</ref><sup>pg 36</sup> of a complex vector space <math>V</math> <math display="block">\operatorname{Hom}_{\overline{\Complex}}(V,\Complex)</math> is a special example because it is isomorphic to the real dual of the underlying real vector space of <math>V,</math> <math>\text{Hom}_\R(V,\R).</math> This is given by the map sending an anti-linear map <math display="block">\ell: V \to \Complex</math>to <math display="block">\operatorname{Im}(\ell) : V \to \R</math> In the other direction, there is the inverse map sending a real dual vector <math display="block">\lambda : V \to \R</math> to <math display="block">\ell(v) = -\lambda(iv) + i\lambda(v)</math> giving the desired map.