Conjugate homogeneous additive map
In mathematics , a function f : V → W {\displaystyle f:V\to W} between two complex vector spaces is said to be antilinear or conjugate-linear if
f ( x + y ) = f ( x ) + f ( y ) (additivity) f ( s x ) = s ¯ f ( x ) (conjugate homogeneity) {\displaystyle {\begin{alignedat}{9}f(x+y)&=f(x)+f(y)&&\qquad {\text{ (additivity) }}\\f(sx)&={\overline {s}}f(x)&&\qquad {\text{ (conjugate homogeneity) }}\\\end{alignedat}}} hold for all vectors
x , y ∈ V {\displaystyle x,y\in V} and every
complex number s , {\displaystyle s,} where
s ¯ {\displaystyle {\overline {s}}} denotes the
complex conjugate of
s . {\displaystyle s.} 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 characterizations [ edit ] A function is called antilinear or conjugate linear if it is additive and conjugate homogeneous . An antilinear functional on a vector space V {\displaystyle V} is a scalar-valued antilinear map.
A function f {\displaystyle f} is called additive if
f ( x + y ) = f ( x ) + f ( y ) for all vectors x , y {\displaystyle f(x+y)=f(x)+f(y)\quad {\text{ for all vectors }}x,y} while it is called
conjugate homogeneous if
f ( a x ) = a ¯ f ( x ) for all vectors x and all scalars a . {\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
f {\displaystyle f} is called
homogeneous if
f ( a x ) = a f ( x ) for all vectors x and all scalars a . {\displaystyle f(ax)=af(x)\quad {\text{ for all vectors }}x{\text{ and all scalars }}a.} An antilinear map f : V → W {\displaystyle f:V\to W} may be equivalently described in terms of the linear map f ¯ : V → W ¯ {\displaystyle {\overline {f}}:V\to {\overline {W}}} from V {\displaystyle V} to the complex conjugate vector space W ¯ . {\displaystyle {\overline {W}}.}
Examples [ edit ] Anti-linear dual map [ edit ] Given a complex vector space V {\displaystyle V} of rank 1, we can construct an anti-linear dual map which is an anti-linear map
l : V → C {\displaystyle l:V\to \mathbb {C} } sending an element
x 1 + i y 1 {\displaystyle x_{1}+iy_{1}} for
x 1 , y 1 ∈ R {\displaystyle x_{1},y_{1}\in \mathbb {R} } to
x 1 + i y 1 ↦ a 1 x 1 − i b 1 y 1 {\displaystyle x_{1}+iy_{1}\mapsto a_{1}x_{1}-ib_{1}y_{1}} for some fixed real numbers
a 1 , b 1 . {\displaystyle 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 , … , e n {\displaystyle e_{1},\ldots ,e_{n}} and each standard basis element as
e k = x k + i y k {\displaystyle e_{k}=x_{k}+iy_{k}} then an anti-linear complex map to
C {\displaystyle \mathbb {C} } will be of the form
∑ k x k + i y k ↦ ∑ k a k x k − i b k y k {\displaystyle \sum _{k}x_{k}+iy_{k}\mapsto \sum _{k}a_{k}x_{k}-ib_{k}y_{k}} for
a k , b k ∈ R . {\displaystyle a_{k},b_{k}\in \mathbb {R} .} Isomorphism of anti-linear dual with real dual [ edit ] The anti-linear dual[1] pg 36 of a complex vector space V {\displaystyle V}
Hom C ¯ ( V , C ) {\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
V , {\displaystyle V,} Hom R ( V , R ) . {\displaystyle {\text{Hom}}_{\mathbb {R} }(V,\mathbb {R} ).} This is given by the map sending an anti-linear map
ℓ : V → C {\displaystyle \ell :V\to \mathbb {C} } to
Im ( ℓ ) : V → R {\displaystyle \operatorname {Im} (\ell ):V\to \mathbb {R} } In the other direction, there is the inverse map sending a real dual vector
λ : V → R {\displaystyle \lambda :V\to \mathbb {R} } to
ℓ ( v ) = − λ ( i v ) + i λ ( v ) {\displaystyle \ell (v)=-\lambda (iv)+i\lambda (v)} giving the desired map.
Properties [ edit ] The composite of two antilinear maps is a linear map . The class of semilinear maps generalizes the class of antilinear maps.
Anti-dual space [ edit ] The vector space of all antilinear forms on a vector space X {\displaystyle X} is called the algebraic anti-dual space of X . {\displaystyle X.} If X {\displaystyle X} is a topological vector space , then the vector space of all continuous antilinear functionals on X , {\displaystyle X,} denoted by X ¯ ′ , {\textstyle {\overline {X}}^{\prime },} is called the continuous anti-dual space or simply the anti-dual space of X {\displaystyle X} if no confusion can arise.
When H {\displaystyle H} is a normed space then the canonical norm on the (continuous) anti-dual space X ¯ ′ , {\textstyle {\overline {X}}^{\prime },} denoted by ‖ f ‖ X ¯ ′ , {\textstyle \|f\|_{{\overline {X}}^{\prime }},} is defined by using this same equation:
‖ f ‖ X ¯ ′ := sup ‖ x ‖ ≤ 1 , x ∈ X | f ( x ) | for every f ∈ X ¯ ′ . {\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 X ′ {\displaystyle X^{\prime }} of X , {\displaystyle X,} which is defined by
‖ f ‖ X ′ := sup ‖ x ‖ ≤ 1 , x ∈ X | f ( x ) | for every f ∈ X ′ . {\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 f ¯ {\displaystyle {\overline {f}}} of a functional f {\displaystyle f} is defined by sending x ∈ domain f {\displaystyle x\in \operatorname {domain} f} to f ( x ) ¯ . {\textstyle {\overline {f(x)}}.} It satisfies
‖ f ‖ X ′ = ‖ f ¯ ‖ X ¯ ′ and ‖ g ¯ ‖ X ′ = ‖ g ‖ X ¯ ′ {\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
f ∈ X ′ {\displaystyle f\in X^{\prime }} and every
g ∈ X ¯ ′ . {\textstyle g\in {\overline {X}}^{\prime }.} This says exactly that the canonical antilinear
bijection defined by
Cong : X ′ → X ¯ ′ where Cong ( f ) := f ¯ {\displaystyle \operatorname {Cong} ~:~X^{\prime }\to {\overline {X}}^{\prime }\quad {\text{ where }}\quad \operatorname {Cong} (f):={\overline {f}}} as well as its inverse
Cong − 1 : X ¯ ′ → X ′ {\displaystyle \operatorname {Cong} ^{-1}~:~{\overline {X}}^{\prime }\to X^{\prime }} are antilinear
isometries and consequently also
homeomorphisms .
If F = R {\displaystyle \mathbb {F} =\mathbb {R} } then X ′ = X ¯ ′ {\displaystyle X^{\prime }={\overline {X}}^{\prime }} and this canonical map Cong : X ′ → X ¯ ′ {\displaystyle \operatorname {Cong} :X^{\prime }\to {\overline {X}}^{\prime }} reduces down to the identity map.
Inner product spaces
If X {\displaystyle X} is an inner product space then both the canonical norm on X ′ {\displaystyle X^{\prime }} and on X ¯ ′ {\displaystyle {\overline {X}}^{\prime }} satisfies the parallelogram law , which means that the polarization identity can be used to define a canonical inner product on X ′ {\displaystyle X^{\prime }} and also on X ¯ ′ , {\displaystyle {\overline {X}}^{\prime },} which this article will denote by the notations
⟨ f , g ⟩ X ′ := ⟨ g ∣ f ⟩ X ′ and ⟨ f , g ⟩ X ¯ ′ := ⟨ g ∣ f ⟩ X ¯ ′ {\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
X ′ {\displaystyle X^{\prime }} and
X ¯ ′ {\displaystyle {\overline {X}}^{\prime }} into Hilbert spaces. The inner products
⟨ f , g ⟩ X ′ {\textstyle \langle f,g\rangle _{X^{\prime }}} and
⟨ f , g ⟩ X ¯ ′ {\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
f ↦ ⟨ f , f ⟩ X ′ {\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
f ∈ X ′ : {\displaystyle f\in X^{\prime }:} sup ‖ x ‖ ≤ 1 , x ∈ X | f ( x ) | = ‖ f ‖ X ′ = ⟨ f , f ⟩ X ′ = ⟨ f ∣ f ⟩ X ′ . {\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 X {\displaystyle X} is an inner product space then the inner products on the dual space X ′ {\displaystyle X^{\prime }} and the anti-dual space X ¯ ′ , {\textstyle {\overline {X}}^{\prime },} denoted respectively by ⟨ ⋅ , ⋅ ⟩ X ′ {\textstyle \langle \,\cdot \,,\,\cdot \,\rangle _{X^{\prime }}} and ⟨ ⋅ , ⋅ ⟩ X ¯ ′ , {\textstyle \langle \,\cdot \,,\,\cdot \,\rangle _{{\overline {X}}^{\prime }},} are related by
⟨ f ¯ | g ¯ ⟩ X ¯ ′ = ⟨ f | g ⟩ X ′ ¯ = ⟨ g | f ⟩ X ′ for all f , g ∈ X ′ {\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
⟨ f ¯ | g ¯ ⟩ X ′ = ⟨ f | g ⟩ X ¯ ′ ¯ = ⟨ g | f ⟩ X ¯ ′ for all f , g ∈ X ¯ ′ . {\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 }.} See also [ edit ] Citations [ edit ]
References [ edit ] 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 .