Antilinear map

A function is called antilinear or conjugate linear if it is additive and conjugate homogeneous. An antilinear functional on a vector space ${\displaystyle V}$ is a scalar-valued antilinear map.

A function ${\displaystyle f}$ is called additive if

$${\displaystyle f(x+y)=f(x)+f(y)\quad {\text{ for all vectors }}x,y}$$

while it is called conjugate homogeneous if

$${\displaystyle f(ax)={\overline {a}}f(x)\quad {\text{ for all vectors }}x{\text{ and all scalars }}a.}$$

In contrast, a linear map is a function that is additive and homogeneous, where ${\displaystyle f}$ is called homogeneous if

$${\displaystyle f(ax)=af(x)\quad {\text{ for all vectors }}x{\text{ and all scalars }}a.}$$

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

Anti-linear dual map

Given a complex vector space ${\displaystyle V}$ of rank 1, we can construct an anti-linear dual map which is an anti-linear map

$${\displaystyle l:V\to \mathbb {C} }$$

sending an element ${\displaystyle x_{1}+iy_{1}}$ for ${\displaystyle x_{1},y_{1}\in \mathbb {R} }$ to

$${\displaystyle x_{1}+iy_{1}\mapsto a_{1}x_{1}-ib_{1}y_{1}}$$

for some fixed real numbers ${\displaystyle a_{1},b_{1}.}$ We can extend this to any finite dimensional complex vector space, where if we write out the standard basis ${\displaystyle e_{1},\ldots ,e_{n}}$ and each standard basis element as

$${\displaystyle e_{k}=x_{k}+iy_{k}}$$

then an anti-linear complex map to ${\displaystyle \mathbb {C} }$ will be of the form

$${\displaystyle \sum _{k}x_{k}+iy_{k}\mapsto \sum _{k}a_{k}x_{k}-ib_{k}y_{k}}$$

for ${\displaystyle a_{k},b_{k}\in \mathbb {R} .}$

Isomorphism of anti-linear dual with real dual

The anti-linear dual[1]^{pg 36} of a complex vector space ${\displaystyle V}$

$${\displaystyle \operatorname {Hom} _{\overline {\mathbb {C} }}(V,\mathbb {C} )}$$

is a special example because it is isomorphic to the real dual of the underlying real vector space of ${\displaystyle V,}$ ${\displaystyle {\text{Hom}}_{\mathbb {R} }(V,\mathbb {R} ).}$ This is given by the map sending an anti-linear map

$${\displaystyle \ell :V\to \mathbb {C} }$$

to

$${\displaystyle \operatorname {Im} (\ell ):V\to \mathbb {R} }$$

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

$${\displaystyle \lambda :V\to \mathbb {R} }$$

to

$${\displaystyle \ell (v)=-\lambda (iv)+i\lambda (v)}$$

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.

The vector space of all antilinear forms on a vector space ${\displaystyle X}$ is called the algebraic anti-dual space of ${\displaystyle X.}$ If ${\displaystyle X}$ is a topological vector space, then the vector space of all continuous antilinear functionals on ${\displaystyle X,}$ denoted by ${\textstyle {\overline {X}}^{\prime },}$ is called the continuous anti-dual space or simply the anti-dual space of ${\displaystyle X}$[2] if no confusion can arise.

When ${\displaystyle H}$ is a normed space then the canonical norm on the (continuous) anti-dual space ${\textstyle {\overline {X}}^{\prime },}$ denoted by ${\textstyle \|f\|_{{\overline {X}}^{\prime }},}$ is defined by using this same equation:[2]

$${\displaystyle \|f\|_{{\overline {X}}^{\prime }}~:=~\sup _{\|x\|\leq 1,x\in X}|f(x)|\quad {\text{ for every }}f\in {\overline {X}}^{\prime }.}$$

This formula is identical to the formula for the dual norm on the continuous dual space ${\displaystyle X^{\prime }}$ of ${\displaystyle X,}$ which is defined by[2]

$${\displaystyle \|f\|_{X^{\prime }}~:=~\sup _{\|x\|\leq 1,x\in X}|f(x)|\quad {\text{ for every }}f\in X^{\prime }.}$$

Canonical isometry between the dual and anti-dual

The complex conjugate ${\displaystyle {\overline {f}}}$ of a functional ${\displaystyle f}$ is defined by sending ${\displaystyle x\in \operatorname {domain} f}$ to ${\textstyle {\overline {f(x)}}.}$ It satisfies

$${\displaystyle \|f\|_{X^{\prime }}~=~\left\|{\overline {f}}\right\|_{{\overline {X}}^{\prime }}\quad {\text{ and }}\quad \left\|{\overline {g}}\right\|_{X^{\prime }}~=~\|g\|_{{\overline {X}}^{\prime }}}$$

for every ${\displaystyle f\in X^{\prime }}$ and every ${\textstyle g\in {\overline {X}}^{\prime }.}$ This says exactly that the canonical antilinear bijection defined by

$${\displaystyle \operatorname {Cong} ~:~X^{\prime }\to {\overline {X}}^{\prime }\quad {\text{ where }}\quad \operatorname {Cong} (f):={\overline {f}}}$$

as well as its inverse ${\displaystyle \operatorname {Cong} ^{-1}~:~{\overline {X}}^{\prime }\to X^{\prime }}$ are antilinear isometries and consequently also homeomorphisms.

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

Inner product spaces

If ${\displaystyle X}$ is an inner product space then both the canonical norm on ${\displaystyle X^{\prime }}$ and on ${\displaystyle {\overline {X}}^{\prime }}$ satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on ${\displaystyle X^{\prime }}$ and also on ${\displaystyle {\overline {X}}^{\prime },}$ which this article will denote by the notations

$${\displaystyle \langle f,g\rangle _{X^{\prime }}:=\langle g\mid f\rangle _{X^{\prime }}\quad {\text{ and }}\quad \langle f,g\rangle _{{\overline {X}}^{\prime }}:=\langle g\mid f\rangle _{{\overline {X}}^{\prime }}}$$

where this inner product makes ${\displaystyle X^{\prime }}$ and ${\displaystyle {\overline {X}}^{\prime }}$ into Hilbert spaces. The inner products ${\textstyle \langle f,g\rangle _{X^{\prime }}}$ and ${\textstyle \langle f,g\rangle _{{\overline {X}}^{\prime }}}$ are antilinear in their second arguments. Moreover, the canonical norm induced by this inner product (that is, the norm defined by ${\textstyle f\mapsto {\sqrt {\left\langle f,f\right\rangle _{X^{\prime }}}}}$) 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 ${\displaystyle f\in X^{\prime }:}$

$${\displaystyle \sup _{\|x\|\leq 1,x\in X}|f(x)|=\|f\|_{X^{\prime }}~=~{\sqrt {\langle f,f\rangle _{X^{\prime }}}}~=~{\sqrt {\langle f\mid f\rangle _{X^{\prime }}}}.}$$

If ${\displaystyle X}$ is an inner product space then the inner products on the dual space ${\displaystyle X^{\prime }}$ and the anti-dual space ${\textstyle {\overline {X}}^{\prime },}$ denoted respectively by ${\textstyle \langle \,\cdot \,,\,\cdot \,\rangle _{X^{\prime }}}$ and ${\textstyle \langle \,\cdot \,,\,\cdot \,\rangle _{{\overline {X}}^{\prime }},}$ are related by

$${\displaystyle \langle \,{\overline {f}}\,|\,{\overline {g}}\,\rangle _{{\overline {X}}^{\prime }}={\overline {\langle \,f\,|\,g\,\rangle _{X^{\prime }}}}=\langle \,g\,|\,f\,\rangle _{X^{\prime }}\qquad {\text{ for all }}f,g\in X^{\prime }}$$

and

$${\displaystyle \langle \,{\overline {f}}\,|\,{\overline {g}}\,\rangle _{X^{\prime }}={\overline {\langle \,f\,|\,g\,\rangle _{{\overline {X}}^{\prime }}}}=\langle \,g\,|\,f\,\rangle _{{\overline {X}}^{\prime }}\qquad {\text{ for all }}f,g\in {\overline {X}}^{\prime }.}$$

• Cauchy's functional equation – Functional equation
• Complex conjugate – Fundamental operation on complex numbers
• Complex conjugate vector space – Mathematics conceptPages displaying short descriptions of redirect targets
• Fundamental theorem of Hilbert spaces
• Inner product space – Generalization of the dot product; used to define Hilbert spaces
• Linear map – Mathematical function, in linear algebra
• Matrix consimilarity
• Riesz representation theorem – Theorem about the dual of a Hilbert space
• Sesquilinear form – Generalization of a bilinear form
• Time reversal – Time reversal symmetry in physics

• 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.