In mathematics, a function between two complex vector spaces is said to be antilinear or conjugate-linear and every complex number denotes the complex conjugate

Antilinear maps stand in contrast to linear mapss, 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 productss and Hilbert spaces.

Definitions and characterizations

A function is called or if it is additive and conjugate homogeneous. An is a scalar-valued antilinear map.

is called while it is called In contrast, a linear map is a function that is additive and homogeneous is called

may be equivalently described in terms of the linear map to the complex conjugate vector space

Examples

Anti-linear dual map

We can extend this to any finite dimensional complex vector space, where if we write out the standard basis

Isomorphism of anti-linear dual with real dual

The anti-linear dualpg 36 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

is called the is a topological vector space, then the vector space of all is called the or simply the if no confusion can arise.

is a normed space is defined by using this same equation:

This formula is identical to the formula for the on the continuous dual space which is defined by

Canonical isometry between the dual and anti-dual

The complex conjugate This says exactly that the canonical antilinear bijection are antilinear isometries and consequently also homeomorphisms.

reduces down to the identity map.

Inner product spaces

is an inner product space satisfies the parallelogram law, which means that the polarization identity can be used to define a ) is consistent with the dual norm (that is, as defined above by the supremum

is an inner product space

See also

Citations

References

• Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. . (antilinear maps are discussed in section 3.3).

• Horn and Johnson, Matrix Analysis, Cambridge University Press, 1985. . (antilinear maps are discussed in section 4.6).