Complex Conjugation
The complexified vector space VC has more structure than an ordinary complex vector space. It comes with a canonical complex conjugation map:
defined by
The map χ may either be regarded as a conjugate-linear map from VC to itself or as a complex linear isomorphism from VC to its complex conjugate .
Conversely, given a complex vector space W with a complex conjugation χ, W is isomorphic as a complex vector space to the complexification VC of the real subspace
In other words, all complex vector spaces with complex conjugation are the complexification of a real vector space.
For example, when W = Cn with the standard complex conjugation
the invariant subspace V is just the real subspace Rn.
Read more about this topic: Complexification
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