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 complexinner products and Hilbert spaces.
A function is called antilinear or conjugate linear if it is additive and conjugate homogeneous. An antilinear functional on a vector space is a scalar-valued antilinear map.
Given a complex vector space of rank 1, we can construct an anti-linear dual map which is an anti-linear map
sending an element for to
for some fixed real numbers We can extend this to any finite dimensional complex vector space, where if we write out the standard basis and each standard basis element as
then an anti-linear complex map to will be of the form
for
Isomorphism of anti-linear dual with real dualedit
The anti-linear dual[1]pg 36 of a complex vector space
is a special example because it is isomorphic to the real dual of the underlying real vector space of This is given by the map sending an anti-linear map
to
In the other direction, there is the inverse map sending a real dual vector
The vector space of all antilinear forms on a vector space is called the algebraic anti-dual space of If is a topological vector space, then the vector space of all continuous antilinear functionals on denoted by is called the continuous anti-dual space or simply the anti-dual space of [2] if no confusion can arise.
When is a normed space then the canonical norm on the (continuous) anti-dual space denoted by is defined by using this same equation:[2]
If then and this canonical map reduces down to the identity map.
Inner product spaces
If is an inner product space then both the canonical norm on and on satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on and also on which this article will denote by the notations
where this inner product makes and into Hilbert spaces. The inner products and are antilinear in their second arguments. Moreover, the canonical norm induced by this inner product (that is, the norm defined by ) 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
If is an inner product space then the inner products on the dual space and the anti-dual space denoted respectively by and are related by
^Birkenhake, Christina (2004). Complex Abelian Varieties. Herbert Lange (Second, augmented ed.). Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN978-3-662-06307-1. OCLC 851380558.
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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 iff x y f x f y additivity f sx s f x conjugate homogeneity displaystyle begin alignedat 9 f x y amp f x f y amp amp qquad text additivity f sx amp overline s f x amp amp 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 Contents 1 Definitions and characterizations 1 1 Examples 1 1 1 Anti linear dual map 1 1 2 Isomorphism of anti linear dual with real dual 2 Properties 3 Anti dual space 4 See also 5 Citations 6 ReferencesDefinitions and characterizations editA function is called antilinear or conjugate linear if it is additive and conjugate homogeneous An antilinear functional on a vector space V displaystyle V nbsp is a scalar valued antilinear map A function f displaystyle f nbsp is called additive iff 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 nbsp while it is called conjugate homogeneous if f ax 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 nbsp In contrast a linear map is a function that is additive and homogeneous where f displaystyle f nbsp is called homogeneous if f ax af 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 nbsp An antilinear map f V W displaystyle f V to W nbsp may be equivalently described in terms of the linear map f V W displaystyle overline f V to overline W nbsp from V displaystyle V nbsp to the complex conjugate vector space W displaystyle overline W nbsp Examples edit Anti linear dual map edit Given a complex vector space V displaystyle V nbsp of rank 1 we can construct an anti linear dual map which is an anti linear mapl V C displaystyle l V to mathbb C nbsp sending an element x1 iy1 displaystyle x 1 iy 1 nbsp for x1 y1 R displaystyle x 1 y 1 in mathbb R nbsp to x1 iy1 a1x1 ib1y1 displaystyle x 1 iy 1 mapsto a 1 x 1 ib 1 y 1 nbsp for some fixed real numbers a1 b1 displaystyle a 1 b 1 nbsp We can extend this to any finite dimensional complex vector space where if we write out the standard basis e1 en displaystyle e 1 ldots e n nbsp and each standard basis element as ek xk iyk displaystyle e k x k iy k nbsp then an anti linear complex map to C displaystyle mathbb C nbsp will be of the form kxk iyk kakxk ibkyk displaystyle sum k x k iy k mapsto sum k a k x k ib k y k nbsp for ak bk R displaystyle a k b k in mathbb R nbsp Isomorphism of anti linear dual with real dual edit The anti linear dual 1 pg 36 of a complex vector space V displaystyle V nbsp HomC V C displaystyle operatorname Hom overline mathbb C V mathbb C nbsp is a special example because it is isomorphic to the real dual of the underlying real vector space of V displaystyle V nbsp HomR V R displaystyle text Hom mathbb R V mathbb R nbsp This is given by the map sending an anti linear map ℓ V C displaystyle ell V to mathbb C nbsp to Im ℓ V R displaystyle operatorname Im ell V to mathbb R nbsp In the other direction there is the inverse map sending a real dual vector l V R displaystyle lambda V to mathbb R nbsp to ℓ v l iv il v displaystyle ell v lambda iv i lambda v nbsp giving the desired map Properties editThe composite of two antilinear maps is a linear map The class of semilinear maps generalizes the class of antilinear maps Anti dual space editThe vector space of all antilinear forms on a vector space X displaystyle X nbsp is called the algebraic anti dual space of X displaystyle X nbsp If X displaystyle X nbsp is a topological vector space then the vector space of all continuous antilinear functionals on X displaystyle X nbsp denoted by X textstyle overline X prime nbsp is called the continuous anti dual space or simply the anti dual space of X displaystyle X nbsp 2 if no confusion can arise When H displaystyle H nbsp is a normed space then the canonical norm on the continuous anti dual space X textstyle overline X prime nbsp denoted by f X textstyle f overline X prime nbsp is defined by using this same equation 2 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 nbsp This formula is identical to the formula for the dual norm on the continuous dual space X displaystyle X prime nbsp of X displaystyle X nbsp which is defined by 2 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 nbsp Canonical isometry between the dual and anti dualThe complex conjugate f displaystyle overline f nbsp of a functional f displaystyle f nbsp is defined by sending x domain f displaystyle x in operatorname domain f nbsp to f x textstyle overline f x nbsp 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 nbsp for every f X displaystyle f in X prime nbsp and every g X textstyle g in overline X prime nbsp 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 nbsp as well as its inverse Cong 1 X X displaystyle operatorname Cong 1 overline X prime to X prime nbsp are antilinear isometries and consequently also homeomorphisms If F R displaystyle mathbb F mathbb R nbsp then X X displaystyle X prime overline X prime nbsp and this canonical map Cong X X displaystyle operatorname Cong X prime to overline X prime nbsp reduces down to the identity map Inner product spacesIf X displaystyle X nbsp is an inner product space then both the canonical norm on X displaystyle X prime nbsp and on X displaystyle overline X prime nbsp satisfies the parallelogram law which means that the polarization identity can be used to define a canonical inner product on X displaystyle X prime nbsp and also on X displaystyle overline X prime nbsp 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 nbsp where this inner product makes X displaystyle X prime nbsp and X displaystyle overline X prime nbsp into Hilbert spaces The inner products f g X textstyle langle f g rangle X prime nbsp and f g X textstyle langle f g rangle overline X prime nbsp 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 nbsp 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 nbsp 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 nbsp If X displaystyle X nbsp is an inner product space then the inner products on the dual space X displaystyle X prime nbsp and the anti dual space X textstyle overline X prime nbsp denoted respectively by X textstyle langle cdot cdot rangle X prime nbsp and X textstyle langle cdot cdot rangle overline X prime nbsp 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 nbsp 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 nbsp See also editCauchy 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 physicsCitations edit 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 b c Treves 2006 pp 112 123 References editBudinich 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 Treves Francois 2006 1967 Topological Vector Spaces Distributions and Kernels Mineola N Y Dover Publications ISBN 978 0 486 45352 1 OCLC 853623322 nbsp This linear algebra related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Antilinear map amp oldid 1189613321, wikipedia, wiki, book, books, library,