Antilinear Map

In mathematics, a mapping 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

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

An antilinear map may be equivalently described in terms of the linear map from to the complex conjugate vector space .

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.