Editing Linear complex structure (section)

== Extension to related vector spaces ==

Let ''V'' be a real vector space with a complex structure ''J''. The [[dual space]] ''V''* has a natural complex structure ''J''* given by the dual (or [[transpose]]) of ''J''. The complexification of the dual space (''V''*)<sup>'''C'''</sup> therefore has a natural decomposition

:<math>(V^*)^\mathbb{C} = (V^*)^{+}\oplus (V^*)^-</math>

into the ±''i'' eigenspaces of ''J''*. Under the natural identification of (''V''*)<sup>'''C'''</sup> with (''V''<sup>'''C'''</sup>)* one can characterize (''V''*)<sup>+</sup> as those complex linear functionals which vanish on ''V''<sup>−</sup>. Likewise (''V''*)<sup>−</sup> consists of those complex linear functionals which vanish on ''V''<sup>+</sup>.

The (complex) [[tensor algebra|tensor]], [[symmetric algebra|symmetric]], and [[exterior algebra]]s over ''V''<sup>'''C'''</sup> also admit decompositions. The exterior algebra is perhaps the most important application of this decomposition. In general, if a vector space ''U'' admits a decomposition ''U'' = ''S'' ⊕ ''T'', then the exterior powers of ''U'' can be decomposed as follows:
:<math>\Lambda^r U = \bigoplus_{p+q=r}(\Lambda^p S)\otimes(\Lambda^q T).</math>

A complex structure ''J'' on ''V'' therefore induces a decomposition
:<math>\Lambda^r\,V^\mathbb{C} = \bigoplus_{p+q=r} \Lambda^{p,q}\,V_J</math>
where
:<math>\Lambda^{p,q}\,V_J\;\stackrel{\mathrm{def}}{=}\, (\Lambda^p\,V^+)\otimes(\Lambda^q\,V^-).</math>
All exterior powers are taken over the complex numbers. So if ''V''<sub>''J''</sub> has complex dimension ''n'' (real dimension 2''n'') then

:<math>\dim_{\mathbb C}\Lambda^{r}\,V^{\mathbb C} = {2n\choose r}\qquad \dim_{\mathbb C}\Lambda^{p,q}\,V_J = {n \choose p}{n \choose q}.</math>
The dimensions add up correctly as a consequence of [[Vandermonde's identity]].

The space of (''p'',''q'')-forms Λ<sup>''p'',''q''</sup> ''V''<sub>''J''</sub>*  is the space of (complex) [[multilinear form]]s on ''V''<sup>'''C'''</sup> which vanish on homogeneous elements unless ''p'' are from ''V''<sup>+</sup> and ''q'' are from ''V''<sup>−</sup>. It is also possible to regard Λ<sup>''p'',''q''</sup> ''V''<sub>''J''</sub>* as the space of real [[multilinear map]]s from ''V''<sub>''J''</sub> to '''C''' which are complex linear in ''p'' terms and [[conjugate-linear]] in ''q'' terms.

See [[complex differential form]] and [[almost complex manifold]] for applications of these ideas.