Antilinear map

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.