# Antilinear map

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In mathematics, a mapping ${\displaystyle f:V\to W}$ from a complex vector space to another is said to be antilinear (or conjugate-linear) if

${\displaystyle f(ax+by)={\bar {a}}f(x)+{\bar {b}}f(y)}$

for all ${\displaystyle a,\,b\,\in \mathbb {C} }$ and all ${\displaystyle x,\,y\,\in V}$, where ${\displaystyle {\bar {a}}}$ and ${\displaystyle {\bar {b}}}$ are the complex conjugates of ${\displaystyle a}$ and ${\displaystyle b}$ respectively. The composite of two antilinear maps is linear. The class of semilinear maps generalizes the class of antilinear maps.

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

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.

## Anti-dual space

The vector space of all antilinear forms on a vector space X is called the algebraic anti-dual space of X. If X is a topological vector space, then the vector space of all continuous antilinear functionals on X is called the continuous anti-dual space or just the anti-dual space of X.[1]