Antilinear map

In mathematics, a mapping f : V → W from a complex vector space to another is said to be antilinear (or conjugate-linear or semilinear, though the latter term is more general) if

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

for all a, b in C and all x, y in V. The composition of two antilinear maps is complex-linear.

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

See also

• Linear map
• complex conjugate
• sesquilinear form