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Antilinear map
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<h2 id="definitions-and-characterizations">Definitions and characterizations</h2> <p>A function is called antilinear or conjugate linear if it is <a href="/facts/Additive_map/lyCB3yy0">additive</a> and <a href="/facts/Conjugate_homogeneous/wF9XX7s5">conjugate homogeneous</a>. An antilinear functional on a vector space V {\displaystyle V} is a scalar-valued antilinear map. </p><p>A function f {\displaystyle f} is called <a href="/facts/Additive_map/lyCB3yy0">additive</a> 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 <a href="/facts/Conjugate_homogeneity/wF9XX7s5">conjugate homogeneous</a> 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 <a href="/facts/Homogeneous/y84KQJxl">homogeneous</a>, 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.} </p><p>An antilinear map f : V → W {\displaystyle f:V\to W} may be equivalently described in terms of the <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> f ¯ : V → W ¯ {\displaystyle {\overline {f}}:V\to {\overline {W}}} from V {\displaystyle V} to the <a href="/facts/Complex_conjugate_vector_space/trWp8y4A">complex conjugate vector space</a> W ¯ . {\displaystyle {\overline {W}}.} </p> <h3>Examples</h3> <h4>Anti-linear dual map</h4> <p>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} .} </p> <h4>Isomorphism of anti-linear dual with real dual</h4> <p>The anti-linear dual<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>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. </p> <h2 id="properties">Properties</h2> <p>The <a href="/facts/Composition_of_relations/tD8joBlo">composite</a> of two antilinear maps is a <a href="/facts/Linear_map/Q1PBKNVV">linear map</a>. The class of <a href="/facts/Semilinear_map/6C6UGyNj">semilinear maps</a> generalizes the class of antilinear maps. </p> <h2 id="anti-dual-space">Anti-dual space</h2> <p>The vector space of all antilinear forms on a vector space X {\displaystyle X} is called the algebraic <a href="/facts/Anti-dual_space/GTXzWMKc">anti-dual space</a> of X . {\displaystyle X.} If X {\displaystyle X} is a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a>, 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} <a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> if no confusion can arise. </p><p>When H {\displaystyle H} is a <a href="/facts/Normed_space/rmDljfyE">normed space</a> 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:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> ‖ 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 }.} </p><p>This formula is identical to the formula for the <a href="/facts/Dual_norm/WDbaz3tR">dual norm</a> on the <a href="/facts/Continuous_dual_space/4Uw4Knz1">continuous dual space</a> X ′ {\displaystyle X^{\prime }} of X , {\displaystyle X,} which is defined by<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> ‖ 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 }.} </p><p>Canonical isometry between the dual and anti-dual </p><p>The <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> 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 <a href="/facts/Bijective_map/j4gTuTmW">bijection</a> 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 <a href="/facts/Isometry/K5wX8ah6">isometries</a> and consequently also <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphisms</a>. </p><p>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 <a href="/facts/Identity_function/9AtsP0LC">identity map</a>. </p><p>Inner product spaces </p><p>If X {\displaystyle X} is an <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a> then both the canonical norm on X ′ {\displaystyle X^{\prime }} and on X ¯ ′ {\displaystyle {\overline {X}}^{\prime }} satisfies the <a href="/facts/Parallelogram_law/JI1LwPxd">parallelogram law</a>, which means that the <a href="/facts/Polarization_identity/l3Vr7sC7">polarization identity</a> 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 }}}}.} </p><p>If X {\displaystyle X} is an <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a> 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 }.} </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Cauchy%2527s_functional_equation/PHkY7R3F">Cauchy's functional equation</a> – Functional equation</li> <li><a href="/facts/Complex_conjugate/7TEaoQDe">Complex conjugate</a> – Fundamental operation on complex numbers</li> <li><a href="/facts/Complex_conjugate_vector_space/trWp8y4A">Complex conjugate vector space</a> – Mathematics conceptPages displaying short descriptions of redirect targets</li> <li><a href="/facts/Fundamental_theorem_of_Hilbert_spaces/o41hQyjm">Fundamental theorem of Hilbert spaces</a></li> <li><a href="/facts/Inner_product_space/HoyjElEy">Inner product space</a> – Vector space with generalized dot product</li> <li><a href="/facts/Linear_map/Q1PBKNVV">Linear map</a> – Mathematical function, in linear algebra</li> <li><a href="/facts/Matrix_consimilarity/oLAg7VdF">Matrix consimilarity</a></li> <li><a href="/facts/Riesz_representation_theorem/9CjKDUII">Riesz representation theorem</a> – Theorem about the dual of a Hilbert space</li> <li><a href="/facts/Sesquilinear_form/Kp6qUA8v">Sesquilinear form</a> – Generalization of a bilinear form</li> <li><a href="/facts/T-symmetry/Tgf2LJMc">Time reversal</a> – Time reversal symmetry in physics</li></ul> <h2 id="citations">Citations</h2> <ul><li>Budinich, P. and Trautman, A. <i>The Spinorial Chessboard</i>. Springer-Verlag, 1988. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-19078-3. (antilinear maps are discussed in section 3.3).</li> <li>Horn and Johnson, <i>Matrix Analysis,</i> Cambridge University Press, 1985. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-38632-2. (antilinear maps are discussed in section 4.6).</li> <li><a href="/facts/Fran%25C3%25A7ois_Tr%25C3%25A8ves/cpjQbAov">Trèves, François</a> (2006) [1967]. <i>Topological Vector Spaces, Distributions and Kernels</i>. Mineola, N.Y.: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-45352-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/853623322">853623322</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Birkenhake, Christina (2004). Complex Abelian Varieties. Herbert Lange (Second, augmented ed.). Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-662-06307-1. OCLC 851380558. <a href="978-3-662-06307-1" target="_blank">978-3-662-06307-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Trèves 2006, pp. 112–123. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Trèves 2006, pp. 112–123. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Trèves 2006, pp. 112–123. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>