Complexification - Complex Conjugation

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.