# Antilinear map

In mathematics, a mapping from a complex vector space to another is said to be **antilinear** (or **conjugate-linear**) if

for all and all , where and are the complex conjugates of and respectively. The composite of two antilinear maps is linear. The class of semilinear maps generalizes the class of antilinear maps.

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

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.

## Anti-dual spaceEdit

The vector space of all antilinear forms on a vector space X is called the **algebraic anti-dual space** of X. If X is a topological vector space, then the vector space of all *continuous* antilinear functionals on X is called the **continuous anti-dual space** or just the **anti-dual space** of X.^{[1]}

## See alsoEdit

## ReferencesEdit

- 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.

## See alsoEdit

- Complex conjugate
- Complex conjugate vector space
- Fundamental theorem of Hilbert spaces
- Linear map
- Sesquilinear form
- Matrix consimilarity
- Time reversal

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**^**Trèves 2006, pp. 112-123.