In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if hold for all vectors x, y \in V and every complex number s, where denotes the complex conjugate of 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.

Definitions and characterizations

A function is called or if it is additive and conjugate homogeneous. An on a vector space V is a scalar-valued antilinear map. A function f is called if while it is called if In contrast, a linear map is a function that is additive and homogeneous, where f is called if An antilinear map f : V \to W may be equivalently described in terms of the linear map from V to the complex conjugate vector space

Examples

Anti-linear dual map

Given a complex vector space V of rank 1, we can construct an anti-linear dual map which is an anti-linear map sending an element x_1 + iy_1 for to for some fixed real numbers a_1,b_1. 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 \Complex will be of the form for

Isomorphism of anti-linear dual with real dual

The anti-linear dual pg 36 of a complex vector space V is a special example because it is isomorphic to the real dual of the underlying real vector space of V, 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.

Properties

The composite of two antilinear maps is a linear map. The class of semilinear maps generalizes the class of antilinear maps.

Anti-dual space

The vector space of all antilinear forms on a vector space X is called the of X. If X is a topological vector space, then the vector space of all antilinear functionals on X, denoted by is called the or simply the of X if no confusion can arise. When H is a normed space then the canonical norm on the (continuous) anti-dual space denoted by is defined by using this same equation: This formula is identical to the formula for the on the continuous dual space X^{\prime} of X, which is defined by Canonical isometry between the dual and anti-dual The complex conjugate of a functional f is defined by sending to It satisfies for every and every This says exactly that the canonical antilinear bijection defined by as well as its inverse are antilinear isometries and consequently also homeomorphisms. If then and this canonical map reduces down to the identity map. Inner product spaces If X is an inner product space then both the canonical norm on X^{\prime} and on satisfies the parallelogram law, which means that the polarization identity can be used to define a and also on which this article will denote by the notations where this inner product makes X^{\prime} and into Hilbert spaces. The inner products and are antilinear in their second arguments. Moreover, the canonical norm induced by this inner product (that is, the norm defined by ) is consistent with the dual norm (that is, as defined above by the supremum over the unit ball); explicitly, this means that the following holds for every If X is an inner product space then the inner products on the dual space X^{\prime} and the anti-dual space denoted respectively by and are related by and

Citations