Editing Complex conjugate of a vector space

{{Short description|Mathematics concept}}
In [[mathematics]], the '''complex conjugate''' of a [[complex numbers|complex]] [[vector space]] <math>V\,</math> is a complex vector space <math>\overline V</math> that has the same elements and additive group structure as <math>V,</math> but whose [[scalar multiplication]] involves [[Complex conjugate|conjugation]] of the scalars. In other words, the scalar multiplication of <math>\overline V</math> satisfies
<math display="block">\alpha\,*\, v = {\,\overline{\alpha} \cdot \,v\,}</math>
where <math>*</math> is the scalar multiplication of <math>\overline{V}</math> and <math>\cdot</math> is the scalar multiplication of <math>V.</math>
The letter <math>v</math> stands for a vector in <math>V,</math> <math>\alpha</math> is a complex number, and <math>\overline{\alpha}</math> denotes the [[complex conjugate]] of <math>\alpha.</math><ref name="Schmüdgen2013">{{cite book|author=K. Schmüdgen|title=Unbounded Operator Algebras and Representation Theory|url=https://books.google.com/books?id=Fx3yBwAAQBAJ&pg=PA16|date=11 November 2013|publisher=Birkhäuser|isbn=978-3-0348-7469-4|page=16}}</ref>

More concretely, the complex conjugate vector space is the same underlying {{em|real}} vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate [[linear complex structure]] <math>J</math> (different multiplication by <math>i</math>).

==Motivation==
If <math>V</math> and <math>W</math> are complex vector spaces, a function <math>f : V \to W</math> is [[Antilinear map|antilinear]] if
<math display="block">f(v + w) = f(v) + f(w) \quad \text{ and } \quad f(\alpha v) = \overline{\alpha} \, f(v)</math>
With the use of the conjugate vector space <math>\overline V</math>, an antilinear map <math>f : V \to W</math> can be regarded as an ordinary [[linear map]] of type <math>\overline{V} \to W.</math> The linearity is checked by noting:
<math display="block">f(\alpha * v) = f(\overline{\alpha} \cdot v) =  \overline{\overline{\alpha}} \cdot f(v) = \alpha \cdot f(v)</math>
Conversely, any linear map defined on <math>\overline{V}</math> gives rise to an antilinear map on <math>V.</math>

This is the same underlying principle as in defining the [[opposite ring]] so that a right <math>R</math>-[[Right module|module]] can be regarded as a left <math>R^{op}</math>-module, or that of an [[opposite category]] so that a [[contravariant functor]] <math>C \to D</math> can be regarded as an ordinary functor of type <math>C^{op} \to D.</math>

==Complex conjugation functor==
A linear map <math>f : V \to W\,</math> gives rise to a corresponding  linear map <math>\overline{f} : \overline{V} \to \overline{W}</math> that has the same action as <math>f.</math> Note that <math>\overline f</math> preserves scalar multiplication because
<math display="block">\overline{f}(\alpha * v) = f(\overline{\alpha} \cdot v) = \overline{\alpha} \cdot f(v) = \alpha * \overline{f}(v)</math>
Thus, complex conjugation <math>V \mapsto \overline{V}</math> and <math>f \mapsto\overline f</math> define a [[functor]] from the [[Category theory|category]] of complex vector spaces to itself.

If <math>V</math> and <math>W</math> are finite-dimensional and the map <math>f</math> is described by the complex [[Matrix (mathematics)|matrix]] <math>A</math> with respect to the [[Basis of a vector space|bases]] <math>\mathcal{B}</math> of <math>V</math> and <math>\mathcal{C}</math> of <math>W,</math> then the map <math>\overline{f}</math> is described by the complex conjugate of <math>A</math> with respect to the bases  <math>\overline{\mathcal{B}}</math> of  <math>\overline{V}</math> and <math>\overline{\mathcal{C}}</math> of <math>\overline{W}.</math>

==Structure of the conjugate==
The vector spaces <math>V</math> and <math>\overline{V}</math> have the same [[Dimension of a vector space|dimension]] over the complex numbers and are therefore [[Isomorphism|isomorphic]] as complex vector spaces. However, there is no [[natural isomorphism]] from  <math>V</math> to <math>\overline{V}.</math>

The double conjugate <math>\overline{\overline{V}}</math> is identical to <math>V.</math>

== Complex conjugate of a Hilbert space ==
Given a [[Hilbert space]] <math>\mathcal{H}</math> (either finite or infinite dimensional), its complex conjugate <math>\overline{\mathcal{H}}</math> is the same vector space as its [[continuous dual space]] <math>\mathcal{H}^{\prime}.</math> 
There is one-to-one antilinear correspondence between continuous linear functionals and vectors. 
In other words, any continuous [[linear functional]] on <math>\mathcal{H}</math> is an inner multiplication to some fixed vector, and vice versa.{{citation needed|date=August 2015}}

Thus, the complex conjugate to a vector <math>v,</math> particularly in finite dimension case, may be denoted as <math>v^\dagger</math> (v-dagger, a [[row vector]] that is the [[conjugate transpose]] to a column vector <math>v</math>).
In [[quantum mechanics]], the conjugate to a ''ket&nbsp;vector''&nbsp;<math>\,|\psi\rangle</math> is denoted as <math>\langle\psi|\,</math> – a ''bra vector'' (see [[bra–ket notation]]).

==See also==

* {{annotated link|Antidual space}}
* {{annotated link|Linear complex structure}}
* {{annotated link|Riesz representation theorem}}
* [[conjugate bundle]]

==References==
{{reflist}}

==Further reading==
* Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Springer-Verlag, 1988. {{ISBN|0-387-19078-3}}. (complex conjugate vector spaces are discussed in section 3.3, pag. 26).

[[Category:Linear algebra]]
[[Category:Vectors (mathematics and physics)|Vector space]]