Antilinear map

In mathematics, a mapping formula_1 from a complex vector space to another is said to be antilinear (or conjugate-linear) if
for all formula_3 and all formula_4, where formula_5 and formula_6 are the complex conjugates of formula_7 and formula_8 respectively. The composite of two antilinear maps is linear. The class of semilinear maps generalizes the class of antilinear maps.
An antilinear map formula_1 may be equivalently described in terms of the linear map formula_10 from formula_11 to the complex conjugate vector space formula_12.
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 is called the algebraic anti-dual space of . If is a topological vector space, then the vector space of all "continuous" antilinear functionals on is called the continuous anti-dual space or just the anti-dual space of .