Antilinear Map
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In mathematics, a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
f : V \to W between two
complex vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s is said to be antilinear or conjugate-linear if \begin f(x + y) &= f(x) + f(y) && \qquad \text \\ f(s x) &= \overline f(x) && \qquad \text \\ \end hold for all vectors x, y \in V and every
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, where \overline denotes the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of s. Antilinear maps stand in contrast to
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s, which are
additive map In algebra, an additive map, Z-linear map or additive function is a function f that preserves the addition operation: f(x + y) = f(x) + f(y) for every pair of elements x and y in the domain of f. For example, any linear map is additive. When the ...
s that are homogeneous rather than
conjugate homogeneous In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the '' ...
. If the vector spaces are
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
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 Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
s and Hilbert spaces.


Definitions and characterizations

A function is called or if it is
additive Additive may refer to: Mathematics * Additive function, a function in number theory * Additive map, a function that preserves the addition operation * Additive set-functionn see Sigma additivity * Additive category, a preadditive category with f ...
and
conjugate homogeneous In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the '' ...
. 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) \quad \text x, y while it is called if f(ax) = \overline f(x) \quad \text x \text a. In contrast, a linear map is a function that is additive and homogeneous, where f is called if f(ax) = a f(x) \quad \text x \text a. An antilinear map f : V \to W may be equivalently described in terms of the
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
\overline : V \to \overline from V to the
complex conjugate vector space In mathematics, the complex conjugate of a complex vector space V\, is a complex vector space \overline V, which has the same elements and additive group structure as V, but whose scalar multiplication involves conjugation of the scalars. In other ...
\overline.


Examples


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 \to \Complex sending an element x_1 + iy_1 for x_1,y_1 \in \R to x_1 + iy_1 \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, \ldots, e_n and each standard basis element as e_k = x_k + iy_k then an anti-linear complex map to \Complexwill be of the form \sum_k x_k + iy_k \mapsto \sum_k a_k x_k - i b_k y_k for a_k,b_k \in \R.


Isomorphism of anti-linear dual with real dual

The anti-linear dualpg 36 of a complex vector space V \operatorname_(V,\Complex) is a special example because it is isomorphic to the real dual of the underlying real vector space of V, \text_\R(V,\R). This is given by the map sending an anti-linear map \ell: V \to \Complexto \operatorname(\ell) : V \to \R In the other direction, there is the inverse map sending a real dual vector \lambda : V \to \R to \ell(v) = -\lambda(iv) + i\lambda(v) giving the desired map.


Properties

The
composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ...
of two antilinear maps is a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
. The class of
semilinear map In linear algebra, particularly projective geometry, a semilinear map between vector spaces ''V'' and ''W'' over a field ''K'' is a function that is a linear map "up to a twist", hence ''semi''-linear, where "twist" means "field automorphism of ''K' ...
s generalizes the class of antilinear maps.


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 In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
, then the vector space of all antilinear functionals on X, denoted by \overline^, 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 \overline^, denoted by \, f\, _, is defined by using this same equation: \, f\, _ ~:=~ \sup_ , f(x), \quad \text f \in \overline^. This formula is identical to the formula for the on the
continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
X^ of X, which is defined by \, f\, _ ~:=~ \sup_ , f(x), \quad \text f \in X^. Canonical isometry between the dual and anti-dual The
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
\overline of a functional f is defined by sending x \in \operatorname f to \overline. It satisfies \, f\, _ ~=~ \left\, \overline\right\, _ \quad \text \quad \left\, \overline\right\, _ ~=~ \, g\, _ for every f \in X^ and every g \in \overline^. This says exactly that the canonical antilinear bijection defined by \operatorname ~:~ X^ \to \overline^ \quad \text \quad \operatorname(f) := \overline as well as its inverse \operatorname^ ~:~ \overline^ \to X^ are antilinear
isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
and consequently also
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
s. If \mathbb = \R then X^ = \overline^ and this canonical map \operatorname : X^ \to \overline^ reduces down to the identity map. Inner product spaces If X is an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
then both the canonical norm on X^ and on \overline^ satisfies the
parallelogram law In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the s ...
, which means that the polarization identity can be used to define a and also on \overline^, which this article will denote by the notations \langle f, g \rangle_ := \langle g \mid f \rangle_ \quad \text \quad \langle f, g \rangle_ := \langle g \mid f \rangle_ where this inner product makes X^ and \overline^ into Hilbert spaces. The inner products \langle f, g \rangle_ and \langle f, g \rangle_ are antilinear in their second arguments. Moreover, the canonical norm induced by this inner product (that is, the norm defined by f \mapsto \sqrt) 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 \in X^: \sup_ , f(x), = \, f\, _ ~=~ \sqrt ~=~ \sqrt. If X is an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
then the inner products on the dual space X^ and the anti-dual space \overline^, denoted respectively by \langle \,\cdot\,, \,\cdot\, \rangle_ and \langle \,\cdot\,, \,\cdot\, \rangle_, are related by \langle \,\overline\, , \,\overline\, \rangle_ = \overline = \langle \,g\, , \,f\, \rangle_ \qquad \text f, g \in X^ and \langle \,\overline\, , \,\overline\, \rangle_ = \overline = \langle \,g\, , \,f\, \rangle_ \qquad \text f, g \in \overline^.


See also

* * * * * * * * * *


Citations


References

* 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). * Functions and mappings Linear algebra {{linear-algebra-stub