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 characterizationsEdit

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 mapEdit

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

Isomorphism of anti-linear dual with real dualEdit

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
In the other direction, there is the inverse map sending a real dual vector
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 spaceEdit

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


See alsoEdit


  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.


  • Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. ISBN 0-387-19078-3. (antilinear maps are discussed in section 3.3).
  • Horn and Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-38632-2. (antilinear maps are discussed in section 4.6).
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.