In mathematics, a function ${\displaystyle f:V\to W}$ between two complex vector spaces is said to be antilinear or conjugate-linear if $${\displaystyle {\begin{alignedat}{9}f(x+y)&=f(x)+f(y)&&\qquad {\text{ (additivity) }}\\f(sx)&={\overline {s}}f(x)&&\qquad {\text{ (conjugate homogeneity) }}\\\end{alignedat}}}$$ hold for all vectors ${\displaystyle x,y\in V}$ and every complex number ${\displaystyle s,}$ where ${\displaystyle {\overline {s}}}$ denotes the complex conjugate of ${\displaystyle s.}$

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.

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 $${\displaystyle f(x+y)=f(x)+f(y)\quad {\text{ for all vectors }}x,y}$$ while it is called conjugate homogeneous if $${\displaystyle f(ax)={\overline {a}}f(x)\quad {\text{ for all vectors }}x{\text{ and all scalars }}a.}$$ In contrast, a linear map is a function that is additive and homogeneous, where ${\displaystyle f}$ is called homogeneous if $${\displaystyle f(ax)=af(x)\quad {\text{ for all vectors }}x{\text{ and all scalars }}a.}$$

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 ${\displaystyle {\overline {W}}.}$

Given a complex vector space ${\displaystyle V}$ of rank 1, we can construct an anti-linear dual map which is an anti-linear map $${\displaystyle l:V\to \mathbb {C} }$$ sending an element ${\displaystyle x_{1}+iy_{1}}$ for ${\displaystyle x_{1},y_{1}\in \mathbb {R} }$ to $${\displaystyle x_{1}+iy_{1}\mapsto a_{1}x_{1}-ib_{1}y_{1}}$$ for some fixed real numbers ${\displaystyle a_{1},b_{1}.}$ We can extend this to any finite dimensional complex vector space, where if we write out the standard basis ${\displaystyle e_{1},\ldots ,e_{n}}$ and each standard basis element as $${\displaystyle e_{k}=x_{k}+iy_{k}}$$ then an anti-linear complex map to ${\displaystyle \mathbb {C} }$ will be of the form $${\displaystyle \sum _{k}x_{k}+iy_{k}\mapsto \sum _{k}a_{k}x_{k}-ib_{k}y_{k}}$$ for ${\displaystyle a_{k},b_{k}\in \mathbb {R} .}$

The anti-linear dual^{pg 36} of a complex vector space ${\displaystyle V}$ $${\displaystyle \operatorname {Hom} _{\overline {\mathbb {C} }}(V,\mathbb {C} )}$$ is a special example because it is isomorphic to the real dual of the underlying real vector space of ${\displaystyle V,}$ ${\displaystyle {\text{Hom}}_{\mathbb {R} }(V,\mathbb {R} ).}$ This is given by the map sending an anti-linear map $${\displaystyle \ell :V\to \mathbb {C} }$$to $${\displaystyle \operatorname {Im} (\ell ):V\to \mathbb {R} }$$ In the other direction, there is the inverse map sending a real dual vector $${\displaystyle \lambda :V\to \mathbb {R} }$$ to $${\displaystyle \ell (v)=-\lambda (iv)+i\lambda (v)}$$ giving the desired map.