Antilinear map

In mathematics, a mapping $f:V\to W$ from a complex vector space to another is said to be antilinear (or conjugate-linear) if

$f(ax+by)={\bar {a}}f(x)+{\bar {b}}f(y)$

for all $a,\,b\,\in \mathbb {C}$ and all $x,\,y\,\in V$, where ${\bar {a}}$ and ${\bar {b}}$ are the complex conjugates of $a$ and $b$ respectively. The composite of two antilinear maps is linear. The class of semilinear maps generalizes the class of antilinear maps.

An antilinear map $f:V\to W$ may be equivalently described in terms of the linear map ${\bar {f}}:V\to {\bar {W}}$ from $V$ to the complex conjugate vector space ${\bar {W}}$.

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 space

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.

See also

• Fundamental theorem of Hilbert spaces
• Inner product space – Generalization of the dot product; used to define Hilbert spaces
• Riesz representation theorem – Theorem about the dual of a Hilbert space
• Sesquilinear form – Generalization of a bilinear form

References

• 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 also

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