Antilinear map

In mathematics, a function between two complex vector spaces is said to be antilinear or conjugate-linear if hold for all vectors and every complex number where denotes the complex conjugate of 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. A function is called or if it is additive and conjugate homogeneous. An on a vector space is a scalar-valued antilinear map. A function is called if while it is called if In contrast, a linear map is a function that is additive and homogeneous, where is called if An antilinear map may be equivalently described in terms of the linear map from to the complex conjugate vector space Given a complex vector space of rank 1, we can construct an anti-linear dual map which is an anti-linear map sending an element for to for some fixed real numbers We can extend this to any finite dimensional complex vector space, where if we write out the standard basis and each standard basis element as then an anti-linear complex map to will be of the form for The anti-linear dualpg 36 of a complex vector space is a special example because it is isomorphic to the real dual of the underlying real vector space of This is given by the map sending an anti-linear map to In the other direction, there is the inverse map sending a real dual vector to giving the desired map. The composite of two antilinear maps is a linear map. The class of semilinear maps generalizes the class of antilinear maps. The vector space of all antilinear forms on a vector space is called the of If is a topological vector space, then the vector space of all antilinear functionals on denoted by is called the or simply the of if no confusion can arise.