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 antilinear or conjugate linear if it is additive and conjugate homogeneous. An antilinear functional on a vector space is a scalar-valued antilinear map.

A function is called additive if while it is called conjugate homogeneous if In contrast, a linear map is a function that is additive and homogeneous, where is called homogeneous if

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


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 dual[1]pg 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.

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 denoted by is called the continuous anti-dual space or simply the anti-dual space of [2] if no confusion can arise.

When is a normed space then the canonical norm on the (continuous) anti-dual space denoted by is defined by using this same equation:[2]

This formula is identical to the formula for the dual norm on the continuous dual space of which is defined by[2]

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 canonical inner product on 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 (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 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

See also


  1. ^ Birkenhake, Christina (2004). Complex Abelian Varieties. Herbert Lange (Second, augmented ed.). Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-662-06307-1. OCLC 851380558.
  2. ^ a b c Trèves 2006, pp. 112–123.