Editing Linear complex structure (section)

==Relation to complexifications==
Given any real vector space ''V'' we may define its [[complexification]] by [[extension of scalars]]:
:<math>V^{\mathbb C}=V\otimes_{\mathbb{R}}\mathbb{C}.</math>
This is a complex vector space whose complex dimension is equal to the real dimension of ''V''. It has a canonical [[complex conjugation]] defined by
:<math>\overline{v\otimes z} = v\otimes\bar z</math>

If ''J'' is a complex structure on ''V'', we may extend ''J'' by linearity to ''V''<sup>'''C'''</sup>:
:<math>J(v\otimes z) = J(v)\otimes z.</math>

Since '''C''' is [[algebraically closed]], ''J'' is guaranteed to have [[eigenvalue]]s which satisfy λ<sup>2</sup> = −1, namely λ = ±''i''.  Thus we may write
:<math>V^{\mathbb C}= V^{+}\oplus V^{-}</math>
where ''V''<sup>+</sup> and ''V''<sup>−</sup> are the [[eigenspace]]s of +''i'' and −''i'', respectively. Complex conjugation interchanges ''V''<sup>+</sup> and ''V''<sup>−</sup>. The projection maps onto the ''V''<sup>±</sup> eigenspaces are given by
:<math>\mathcal P^{\pm} = {1\over 2}(1\mp iJ).</math>
So that
:<math>V^{\pm} = \{v\otimes 1 \mp Jv\otimes i: v \in V\}.</math>

There is a natural complex linear isomorphism between ''V''<sub>''J''</sub> and ''V''<sup>+</sup>, so these vector spaces can be considered the same, while ''V''<sup>−</sup> may be regarded as the [[complex conjugate vector space|complex conjugate]] of ''V''<sub>''J''</sub>.

Note that if ''V''<sub>''J''</sub> has complex dimension ''n'' then both ''V''<sup>+</sup> and ''V''<sup>−</sup> have complex dimension ''n'' while ''V''<sup>'''C'''</sup> has complex dimension 2''n''.

Abstractly, if one starts with a complex vector space ''W'' and takes the complexification of the underlying real space, one obtains a space isomorphic to the direct sum of ''W'' and its conjugate:
:<math>W^{\mathbb C} \cong W\oplus \overline{W}.</math>