Antilinear map

 ( Functions And Mappings ) 

In mathematics, a function $f\; :\; V\; \backslash to\; W$ between two complex vector spaces is said to be antilinear or conjugate-linear if hold for all vectors $x,\; y\; \backslash in\; V$ and every complex number $s,$ where $\backslash overline\{s\}$ denotes the complex conjugate of $s.$

Antilinear maps stand in contrast to Linear operator, which are that are Homogeneous map 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 T-symmetry 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 number inner products and .

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Definitions and characterizations A function is called or if it is Additive map and conjugate homogeneous. An on a vector space $V$ is a scalar-valued antilinear map.

A function $f$ is called if $$f(x\; +\; y)\; =\; f(x)\; +\; f(y)\; \backslash quad\; \backslash text\{\; for\; all\; vectors\; \}\; x,\; y$$ while it is called if $$f(ax)\; =\; \backslash overline\{a\}\; f(x)\; \backslash quad\; \backslash text\{\; for\; all\; vectors\; \}\; x\; \backslash text\{\; and\; all\; scalars\; \}\; a.$$ In contrast, a linear map is a function that is additive and homogeneous, where $f$ is called if $$f(ax)\; =\; a\; f(x)\; \backslash quad\; \backslash text\{\; for\; all\; vectors\; \}\; x\; \backslash text\{\; and\; all\; scalars\; \}\; a.$$

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

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Examples

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Anti-linear dual map Given a complex vector space $V$ of rank 1, we can construct an anti-linear dual map which is an anti-linear map $$l:V\; \backslash to\; \backslash Complex$$ sending an element $x\_1\; +\; iy\_1$ for $x\_1,y\_1\; \backslash in\; \backslash R$ to $$x\_1\; +\; iy\_1\; \backslash mapsto\; a\_1\; x\_1\; -\; i\; b\_1\; y\_1$$ for some fixed real numbers $a\_1,b\_1.$ We can extend this to any finite dimensional complex vector space, where if we write out the standard basis $e\_1,\; \backslash ldots,\; e\_n$ and each standard basis element as $$e\_k\; =\; x\_k\; +\; iy\_k$$ then an anti-linear complex map to $\backslash Complex$ will be of the form $$\backslash sum\_k\; x\_k\; +\; iy\_k\; \backslash mapsto\; \backslash sum\_k\; a\_k\; x\_k\; -\; i\; b\_k\; y\_k$$ for $a\_k,b\_k\; \backslash in\; \backslash R.$

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Isomorphism of anti-linear dual with real dual The anti-linear dual

Christina Birkenhake (2025). 9783662063071, Springer Berlin Heidelberg. ISBN 9783662063071

^{pg 36} of a complex vector space $V$ $$\backslash operatorname\{Hom\}\_\{\backslash overline\{\backslash Complex\}\}(V,\backslash Complex)$$ is a special example because it is isomorphic to the real dual of the underlying real vector space of $V,$ $\backslash text\{Hom\}\_\backslash R(V,\backslash R).$ This is given by the map sending an anti-linear map $$\backslash ell:\; V\; \backslash to\; \backslash Complex$$to $$\backslash operatorname\{Im\}(\backslash ell)\; :\; V\; \backslash to\; \backslash R$$ In the other direction, there is the inverse map sending a real dual vector $$\backslash lambda\; :\; V\; \backslash to\; \backslash R$$ to $$\backslash ell(v)\; =\; -\backslash lambda(iv)\; +\; i\backslash lambda(v)$$ giving the desired map.

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Properties The composite of two antilinear maps is a linear map. The class of generalizes the class of antilinear maps by generalizing the field.

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Anti-dual space The vector space of all antilinear forms on a vector space $X$ is called the of $X.$ If $X$ is a topological vector space, then the vector space of all antilinear functionals on $X,$ denoted by $\backslash overline\{X\}^\{\backslash prime\},$ is called the or simply the of $X$ if no confusion can arise.

When $H$ is a normed space then the canonical norm on the (continuous) anti-dual space $\backslash overline\{X\}^\{\backslash prime\},$ denoted by $\backslash |f\backslash |\_\{\backslash overline\{X\}^\{\backslash prime\}\},$ is defined by using this same equation: $$\backslash |f\backslash |\_\{\backslash overline\{X\}^\{\backslash prime\}\}\; ~:=~\; \backslash sup\_\{\backslash |x\backslash |\; \backslash leq\; 1,\; x\; \backslash in\; X\}\; |f(x)|\; \backslash quad\; \backslash text\{\; for\; every\; \}\; f\; \backslash in\; \backslash overline\{X\}^\{\backslash prime\}.$$

This formula is identical to the formula for the on the continuous dual space $X^\{\backslash prime\}$ of $X,$ which is defined by $$\backslash |f\backslash |\_\{X^\{\backslash prime\}\}\; ~:=~\; \backslash sup\_\{\backslash |x\backslash |\; \backslash leq\; 1,\; x\; \backslash in\; X\}\; |f(x)|\; \backslash quad\; \backslash text\{\; for\; every\; \}\; f\; \backslash in\; X^\{\backslash prime\}.$$

Canonical isometry between the dual and anti-dual

The complex conjugate $\backslash overline\{f\}$ of a functional $f$ is defined by sending $x\; \backslash in\; \backslash operatorname\{domain\}\; f$ to $\backslash overline\{f(x)\}.$ It satisfies $$\backslash |f\backslash |\_\{X^\{\backslash prime\}\}\; ~=~\; \backslash left\backslash |\backslash overline\{f\}\backslash right\backslash |\_\{\backslash overline\{X\}^\{\backslash prime\}\}\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash left\backslash |\backslash overline\{g\}\backslash right\backslash |\_\{X^\{\backslash prime\}\}\; ~=~\; \backslash |g\backslash |\_\{\backslash overline\{X\}^\{\backslash prime\}\}$$ for every $f\; \backslash in\; X^\{\backslash prime\}$ and every $g\; \backslash in\; \backslash overline\{X\}^\{\backslash prime\}.$ This says exactly that the canonical antilinear Bijective map defined by $$\backslash operatorname\{Cong\}\; ~:~\; X^\{\backslash prime\}\; \backslash to\; \backslash overline\{X\}^\{\backslash prime\}\; \backslash quad\; \backslash text\{\; where\; \}\; \backslash quad\; \backslash operatorname\{Cong\}(f)\; :=\; \backslash overline\{f\}$$ as well as its inverse $\backslash operatorname\{Cong\}^\{-1\}\; ~:~\; \backslash overline\{X\}^\{\backslash prime\}\; \backslash to\; X^\{\backslash prime\}$ are antilinear Isometry and consequently also .

If $\backslash mathbb\{F\}\; =\; \backslash R$ then $X^\{\backslash prime\}\; =\; \backslash overline\{X\}^\{\backslash prime\}$ and this canonical map $\backslash operatorname\{Cong\}\; :\; X^\{\backslash prime\}\; \backslash to\; \backslash overline\{X\}^\{\backslash prime\}$ reduces down to the identity map.

Inner product spaces

If $X$ is an inner product space then both the canonical norm on $X^\{\backslash prime\}$ and on $\backslash overline\{X\}^\{\backslash prime\}$ satisfies the parallelogram law, which means that the polarization identity can be used to define a and also on $\backslash overline\{X\}^\{\backslash prime\},$ which this article will denote by the notations $$\backslash langle\; f,\; g\; \backslash rangle\_\{X^\{\backslash prime\}\}\; :=\; \backslash langle\; g\; \backslash mid\; f\; \backslash rangle\_\{X^\{\backslash prime\}\}\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash langle\; f,\; g\; \backslash rangle\_\{\backslash overline\{X\}^\{\backslash prime\}\}\; :=\; \backslash langle\; g\; \backslash mid\; f\; \backslash rangle\_\{\backslash overline\{X\}^\{\backslash prime\}\}$$ where this inner product makes $X^\{\backslash prime\}$ and $\backslash overline\{X\}^\{\backslash prime\}$ into Hilbert spaces. The inner products $\backslash langle\; f,\; g\; \backslash rangle\_\{X^\{\backslash prime\}\}$ and $\backslash langle\; f,\; g\; \backslash rangle\_\{\backslash overline\{X\}^\{\backslash prime\}\}$ are antilinear in their second arguments. Moreover, the canonical norm induced by this inner product (that is, the norm defined by $f\; \backslash mapsto\; \backslash sqrt\{\backslash left\backslash langle\; f,\; f\; \backslash right\backslash rangle\_\{X^\{\backslash prime\}\}\}$) 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 $f\; \backslash in\; X^\{\backslash prime\}:$ $$\backslash sup\_\{\backslash |x\backslash |\; \backslash leq\; 1,\; x\; \backslash in\; X\}\; |f(x)|\; =\; \backslash |f\backslash |\_\{X^\{\backslash prime\}\}\; ~=~\; \backslash sqrt\{\backslash langle\; f,\; f\; \backslash rangle\_\{X^\{\backslash prime\}\}\}\; ~=~\; \backslash sqrt\{\backslash langle\; f\; \backslash mid\; f\; \backslash rangle\_\{X^\{\backslash prime\}\}\}.$$

= \langle \,g\, | \,f\, \rangle_{X^{\prime}} \qquad \text{ for all } f, g \in X^{\prime} and = \langle \,g\, | \,f\, \rangle_{\overline{X}^{\prime}} \qquad \text{ for all } f, g \in \overline{X}^{\prime}.

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See also

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Citations

• 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).

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