Editing Complexification (section)

== Complex conjugation ==
The complexified vector space {{math|''V''{{i sup|'''C'''}}}} has more structure than an ordinary complex vector space. It comes with a [[canonical form|canonical]] [[complex conjugation]] map:
:<math>\chi : V^{\Complex} \to \overline{V^{\Complex}}</math>
defined by
:<math>\chi(v\otimes z) = v\otimes \bar z.</math> 
The map {{mvar|χ}} may either be regarded as a [[conjugate-linear map]] from {{math|''V''{{i sup|'''C'''}}}} to itself or as a complex linear [[isomorphism]] from {{math|''V''{{i sup|'''C'''}}}} to its [[complex conjugate vector space|complex conjugate]] <math>\overline {V^{\Complex}}</math>.

Conversely, given a complex vector space {{math|''W''}} with a complex conjugation {{mvar|χ}}, {{math|''W''}} is isomorphic as a complex vector space to the complexification {{math|''V''{{i sup|'''C'''}}}} of the real subspace
:<math>V = \{ w \in W : \chi(w) = w \}.</math>
In other words, all complex vector spaces with complex conjugation are the complexification of a real vector space.

For example, when {{math|1=''W'' = '''C'''<sup>''n''</sup>}} with the standard complex conjugation
:<math>\chi(z_1,\ldots,z_n) = (\bar z_1,\ldots,\bar z_n)</math>
the invariant subspace {{math|''V''}} is just the real subspace {{math|'''R'''<sup>''n''</sup>}}.