In mathematics, a function ${\displaystyle f:V\to W}$ between two complex vector spaces is said to be antilinear or conjugate-linear if

$${\displaystyle {\begin{alignedat}{9}f(x+y)&=f(x)+f(y)&&\qquad {\text{ (additivity) ))\\f(sx)&={\overline {s))f(x)&&\qquad {\text{ (conjugate homogeneity) ))\\\end{alignedat))}$$

${\displaystyle x,y\in V}$

${\displaystyle s,}$

${\displaystyle {\overline {s))}$

${\displaystyle s.}$

Antilinear maps stand in contrast to linear maps, which are additive maps that are homogeneous rather than conjugate homogeneous. If the vector spaces are real then antilinearity is the same as linearity.

Antilinear maps occur in quantum mechanics in the study of time reversal and in 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 inner products and Hilbert spaces.

Definitions and characterizations

A function is called antilinear or conjugate linear if it is additive and conjugate homogeneous. An antilinear functional on a vector space ${\displaystyle V}$ is a scalar-valued antilinear map.

A function ${\displaystyle f}$ is called additive if

$${\displaystyle f(x+y)=f(x)+f(y)\quad {\text{ for all vectors ))x,y}$$

$${\displaystyle f(ax)={\overline {a))f(x)\quad {\text{ for all vectors ))x{\text{ and all scalars ))a.}$$

${\displaystyle f}$

$${\displaystyle f(ax)=af(x)\quad {\text{ for all vectors ))x{\text{ and all scalars ))a.}$$

An antilinear map ${\displaystyle f:V\to W}$ may be equivalently described in terms of the linear map ${\displaystyle {\overline {f)):V\to {\overline {W))}$ from ${\displaystyle V}$ to the complex conjugate vector space ${\displaystyle {\overline {W)).}$

Examples

Anti-linear dual map

Given a complex vector space ${\displaystyle V}$ of rank 1, we can construct an anti-linear dual map which is an anti-linear map

$${\displaystyle l:V\to \mathbb {C} }$$

${\displaystyle x_{1}+iy_{1))$

${\displaystyle x_{1},y_{1}\in \mathbb {R} }$

$${\displaystyle x_{1}+iy_{1}\mapsto a_{1}x_{1}-ib_{1}y_{1))$$

${\displaystyle a_{1},b_{1}.}$

${\displaystyle e_{1},\ldots ,e_{n))$

$${\displaystyle e_{k}=x_{k}+iy_{k))$$

${\displaystyle \mathbb {C} }$

$${\displaystyle \sum _{k}x_{k}+iy_{k}\mapsto \sum _{k}a_{k}x_{k}-ib_{k}y_{k))$$

${\displaystyle a_{k},b_{k}\in \mathbb {R} .}$

Isomorphism of anti-linear dual with real dual

The anti-linear dual^{[1]pg 36} of a complex vector space ${\displaystyle V}$

$${\displaystyle \operatorname {Hom} _{\overline {\mathbb {C} ))(V,\mathbb {C} )}$$

${\displaystyle V,}$

${\displaystyle {\text{Hom))_{\mathbb {R} }(V,\mathbb {R} ).}$

$${\displaystyle \ell :V\to \mathbb {C} }$$

$${\displaystyle \operatorname {Im} (\ell ):V\to \mathbb {R} }$$

$${\displaystyle \lambda :V\to \mathbb {R} }$$

$${\displaystyle \ell (v)=-\lambda (iv)+i\lambda (v)}$$

Properties

The composite of two antilinear maps is a linear map. The class of semilinear maps generalizes the class of antilinear maps.

Anti-dual space

The vector space of all antilinear forms on a vector space ${\displaystyle X}$ is called the algebraic anti-dual space of ${\displaystyle X.}$ If ${\displaystyle X}$ is a topological vector space, then the vector space of all continuous antilinear functionals on ${\displaystyle X,}$ denoted by ${\textstyle {\overline {X))^{\prime },}$ is called the continuous anti-dual space or simply the anti-dual space of ${\displaystyle X}$^[2] if no confusion can arise.

When ${\displaystyle H}$ is a normed space then the canonical norm on the (continuous) anti-dual space ${\textstyle {\overline {X))^{\prime },}$ denoted by ${\textstyle \|f\|_((\overline {X))^{\prime )),}$ is defined by using this same equation:^[2]

$${\displaystyle \|f\|_((\overline {X))^{\prime ))~:=~\sup _{\|x\|\leq 1,x\in X}|f(x)|\quad {\text{ for every ))f\in {\overline {X))^{\prime }.}$$

This formula is identical to the formula for the dual norm on the continuous dual space ${\displaystyle X^{\prime ))$ of ${\displaystyle X,}$ which is defined by^[2]

$${\displaystyle \|f\|_{X^{\prime ))~:=~\sup _{\|x\|\leq 1,x\in X}|f(x)|\quad {\text{ for every ))f\in X^{\prime }.}$$

Canonical isometry between the dual and anti-dual

The complex conjugate ${\displaystyle {\overline {f))}$ of a functional ${\displaystyle f}$ is defined by sending ${\displaystyle x\in \operatorname {domain} f}$ to ${\textstyle {\overline {f(x))).}$ It satisfies

$${\displaystyle \|f\|_{X^{\prime ))~=~\left\|{\overline {f))\right\|_((\overline {X))^{\prime ))\quad {\text{ and ))\quad \left\|{\overline {g))\right\|_{X^{\prime ))~=~\|g\|_((\overline {X))^{\prime ))}$$

${\displaystyle f\in X^{\prime ))$

${\textstyle g\in {\overline {X))^{\prime }.}$

$${\displaystyle \operatorname {Cong} ~:~X^{\prime }\to {\overline {X))^{\prime }\quad {\text{ where ))\quad \operatorname {Cong} (f):={\overline {f))}$$

${\displaystyle \operatorname {Cong} ^{-1}~:~{\overline {X))^{\prime }\to X^{\prime ))$

If ${\displaystyle \mathbb {F} =\mathbb {R} }$ then ${\displaystyle X^{\prime }={\overline {X))^{\prime ))$ and this canonical map ${\displaystyle \operatorname {Cong} :X^{\prime }\to {\overline {X))^{\prime ))$ reduces down to the identity map.

Inner product spaces

If ${\displaystyle X}$ is an inner product space then both the canonical norm on ${\displaystyle X^{\prime ))$ and on ${\displaystyle {\overline {X))^{\prime ))$ satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on ${\displaystyle X^{\prime ))$ and also on ${\displaystyle {\overline {X))^{\prime },}$ which this article will denote by the notations

$${\displaystyle \langle f,g\rangle _{X^{\prime )):=\langle g\mid f\rangle _{X^{\prime ))\quad {\text{ and ))\quad \langle f,g\rangle _((\overline {X))^{\prime )):=\langle g\mid f\rangle _((\overline {X))^{\prime ))}$$

${\displaystyle X^{\prime ))$

${\displaystyle {\overline {X))^{\prime ))$

${\textstyle \langle f,g\rangle _{X^{\prime ))}$

${\textstyle \langle f,g\rangle _((\overline {X))^{\prime ))}$

${\textstyle f\mapsto {\sqrt {\left\langle f,f\right\rangle _{X^{\prime ))))}$

${\displaystyle f\in X^{\prime }:}$

$${\displaystyle \sup _{\|x\|\leq 1,x\in X}|f(x)|=\|f\|_{X^{\prime ))~=~{\sqrt {\langle f,f\rangle _{X^{\prime ))))~=~{\sqrt {\langle f\mid f\rangle _{X^{\prime )))).}$$

If ${\displaystyle X}$ is an inner product space then the inner products on the dual space ${\displaystyle X^{\prime ))$ and the anti-dual space ${\textstyle {\overline {X))^{\prime },}$ denoted respectively by ${\textstyle \langle \,\cdot \,,\,\cdot \,\rangle _{X^{\prime ))}$ and ${\textstyle \langle \,\cdot \,,\,\cdot \,\rangle _((\overline {X))^{\prime )),}$ are related by

$${\displaystyle \langle \,{\overline {f))\,|\,{\overline {g))\,\rangle _((\overline {X))^{\prime ))={\overline {\langle \,f\,|\,g\,\rangle _{X^{\prime ))))=\langle \,g\,|\,f\,\rangle _{X^{\prime ))\qquad {\text{ for all ))f,g\in X^{\prime ))$$

$${\displaystyle \langle \,{\overline {f))\,|\,{\overline {g))\,\rangle _{X^{\prime ))={\overline {\langle \,f\,|\,g\,\rangle _((\overline {X))^{\prime ))))=\langle \,g\,|\,f\,\rangle _((\overline {X))^{\prime ))\qquad {\text{ for all ))f,g\in {\overline {X))^{\prime }.}$$