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.

Definitions and characterizations

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

Examples

Anti-linear dual map

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

Isomorphism of anti-linear dual with real dual

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.

Properties

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

Anti-dual space

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.
When is a normed space then the canonical norm on the 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 of which is defined by
Canonical isometry between the dual and anti-dual
The complex conjugate of a functional 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 is an inner product space then both the canonical norm on 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 and into Hilbert spaces.
The inner products and are antilinear in their second arguments. Moreover, the canonical norm induced by this inner product is consistent with the dual norm ; explicitly, this means that the following holds for every
If is an inner product space then the inner products on the dual space and the anti-dual space denoted respectively by and are related by
and