# Antilinear map

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{{Short description|Conjugate homogeneous additive map}}
In [mathematics](/source/mathematics), a [function](/source/Function_(mathematics)) <math>f : V \to W</math> between two [complex vector space](/source/complex_vector_space)s is said to be '''antilinear''' or '''conjugate-linear''' if 
<math display=block>\begin{alignat}{9}
f(x + y) &= f(x) + f(y) && \qquad \text{ (additivity) } \\
f(s x) &= \overline{s} f(x) && \qquad \text{ (conjugate homogeneity) } \\
\end{alignat}</math>
hold for all vectors <math>x, y \in V</math> and every [complex number](/source/complex_number) <math>s,</math> where <math>\overline{s}</math> denotes the [complex conjugate](/source/complex_conjugate) of <math>s.</math> 

Antilinear maps stand in contrast to [linear map](/source/Linear_operator)s, which are [additive map](/source/additive_map)s that are [homogeneous](/source/Homogeneous_map) rather than [conjugate homogeneous](/source/Conjugate_homogeneity). If the vector spaces are [real](/source/real_vector_space) then antilinearity is the same as linearity.

Antilinear maps occur in [quantum mechanics](/source/quantum_mechanics) in the study of [time reversal](/source/T-symmetry) and in [spinor calculus](/source/spinor_calculus), where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices. Scalar-valued antilinear maps often arise when dealing with [complex](/source/Complex_number) [inner product](/source/Inner_product_space)s and [Hilbert space](/source/Hilbert_space)s. 

== Definitions and characterizations ==

A function is called {{em|antilinear}} or {{em|conjugate linear}} if it is [additive](/source/Additive_map) and [conjugate homogeneous](/source/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](/source/Additive_map)}} 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 homogeneous](/source/Conjugate_homogeneity)}} 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](/source/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](/source/linear_map) <math>\overline{f} : V \to \overline{W}</math> from <math>V</math> to the [complex conjugate vector space](/source/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| 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.

== Properties ==

The [composite](/source/Composition_of_relations) of two antilinear maps is a [linear map](/source/linear_map). The class of [semilinear map](/source/semilinear_map)s generalizes the class of antilinear maps by generalizing the field.

== Anti-dual space ==

The vector space of all antilinear forms on a vector space <math>X</math> is called the {{em|algebraic [anti-dual space](/source/anti-dual_space)}} of <math>X.</math> If <math>X</math> is a [topological vector space](/source/topological_vector_space), then the vector space of all {{em|continuous}} antilinear functionals on <math>X,</math> denoted by <math display="inline">\overline{X}^{\prime},</math> is called the {{em|continuous anti-dual space}} or simply the {{em|anti-dual space}} of <math>X</math>{{sfn|Trèves|2006|pp=112-123}} if no confusion can arise. 

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

This formula is identical to the formula for the {{em|[dual norm](/source/dual_norm)}} on the [continuous dual space](/source/continuous_dual_space) <math>X^{\prime}</math> of <math>X,</math> which is defined by{{sfn|Trèves|2006|pp=112–123}}
<math display=block>\|f\|_{X^{\prime}} ~:=~ \sup_{\|x\| \leq 1, x \in X} |f(x)| \quad \text{ for every } f \in X^{\prime}.</math> 

'''Canonical isometry between the dual and anti-dual'''

The [complex conjugate](/source/complex_conjugate) <math>\overline{f}</math> of a functional <math>f</math> is defined by sending <math>x \in \operatorname{domain} f</math> to <math display="inline">\overline{f(x)}.</math> It satisfies 
<math display=block>\|f\|_{X^{\prime}} ~=~ \left\|\overline{f}\right\|_{\overline{X}^{\prime}} \quad \text{ and } \quad \left\|\overline{g}\right\|_{X^{\prime}} ~=~ \|g\|_{\overline{X}^{\prime}}</math>
for every <math>f \in X^{\prime}</math> and every <math display="inline">g \in \overline{X}^{\prime}.</math> 
This says exactly that the canonical antilinear [bijection](/source/Bijective_map) defined by
<math display=block>\operatorname{Cong} ~:~ X^{\prime} \to \overline{X}^{\prime} \quad \text{ where } \quad \operatorname{Cong}(f) := \overline{f}</math> 
as well as its inverse <math>\operatorname{Cong}^{-1} ~:~ \overline{X}^{\prime} \to X^{\prime}</math> are antilinear [isometries](/source/Isometry) and consequently also [homeomorphism](/source/homeomorphism)s. 

If <math>\mathbb{F} = \R</math> then <math>X^{\prime} = \overline{X}^{\prime}</math> and this canonical map <math>\operatorname{Cong} : X^{\prime} \to \overline{X}^{\prime}</math> reduces down to the [identity map](/source/Identity_function).

'''Inner product spaces'''

If <math>X</math> is an [inner product space](/source/inner_product_space) then both the canonical norm on <math>X^{\prime}</math> and on <math>\overline{X}^{\prime}</math> satisfies the [parallelogram law](/source/parallelogram_law), which means that the [polarization identity](/source/polarization_identity) can be used to define a {{em|canonical inner product on <math>X^{\prime}</math>}} and also on <math>\overline{X}^{\prime},</math> which this article will denote by the notations 
<math display=block>\langle f, g \rangle_{X^{\prime}} := \langle g \mid f \rangle_{X^{\prime}} \quad \text{ and } \quad \langle f, g \rangle_{\overline{X}^{\prime}} := \langle g \mid f \rangle_{\overline{X}^{\prime}}</math> 
where this inner product makes <math>X^{\prime}</math> and <math>\overline{X}^{\prime}</math> into Hilbert spaces. 
The inner products <math display="inline">\langle f, g \rangle_{X^{\prime}}</math> and <math display="inline">\langle f, g \rangle_{\overline{X}^{\prime}}</math> are antilinear in their second arguments. Moreover, the canonical norm induced by this inner product (that is, the norm defined by <math display="inline">f \mapsto \sqrt{\left\langle f, f \right\rangle_{X^{\prime}}}</math>) is consistent with the dual norm (that is, as defined above by the supremum over the unit ball); explicitly, this means that the following holds for every <math>f \in X^{\prime}:</math>
<math display=block>\sup_{\|x\| \leq 1, x \in X} |f(x)| = \|f\|_{X^{\prime}} ~=~ \sqrt{\langle f, f \rangle_{X^{\prime}}} ~=~ \sqrt{\langle f \mid f \rangle_{X^{\prime}}}.</math> 

If <math>X</math> is an [inner product space](/source/inner_product_space) then the inner products on the dual space <math>X^{\prime}</math> and the anti-dual space <math display="inline">\overline{X}^{\prime},</math> denoted respectively by <math display="inline">\langle \,\cdot\,, \,\cdot\, \rangle_{X^{\prime}}</math> and <math display="inline">\langle \,\cdot\,, \,\cdot\, \rangle_{\overline{X}^{\prime}},</math> are related by<math display=block>\langle \,\overline{f}\, | \,\overline{g}\, \rangle_{\overline{X}^{\prime}} 
= \overline{\langle \,f\, | \,g\, \rangle_{X^{\prime}}} 
= \langle \,g\, | \,f\, \rangle_{X^{\prime}} \qquad \text{ for all } f, g \in X^{\prime}</math>
and
<math display=block>\langle \,\overline{f}\, | \,\overline{g}\, \rangle_{X^{\prime}} 
= \overline{\langle \,f\, | \,g\, \rangle_{\overline{X}^{\prime}}} 
= \langle \,g\, | \,f\, \rangle_{\overline{X}^{\prime}} \qquad \text{ for all } f, g \in \overline{X}^{\prime}.</math>

== See also ==

* {{annotated link|Cauchy's functional equation}}
* {{annotated link|Complex conjugate}}
* {{annotated link|Complex conjugate vector space}}
* {{annotated link|Fundamental theorem of Hilbert spaces}}
* {{annotated link|Inner product space}}
* {{annotated link|Linear map}}
* {{annotated link|Matrix consimilarity}}
* {{annotated link|Riesz representation theorem}}
* {{annotated link|Sesquilinear form}}
* {{annotated link|T-symmetry|Time reversal}}

== Citations ==
{{reflist}}
{{reflist|group=note}}

== References ==

* [Budinich, P.](/source/Paolo_Budinich) and [Trautman, A.](/source/Andrzej_Trautman) ''The Spinorial Chessboard''. Springer-Verlag, 1988. {{isbn|0-387-19078-3}}. (antilinear maps are discussed in section 3.3).
* Horn and Johnson, ''Matrix Analysis,'' Cambridge University Press, 1985. {{isbn|0-521-38632-2}}. (antilinear maps are discussed in section 4.6).
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->

Category:Functions and mappings
Category:Linear algebra
Category:Types of functions

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Adapted from the Wikipedia article [Antilinear map](https://en.wikipedia.org/wiki/Antilinear_map) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Antilinear_map?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.