Editing Complexification (section)

==Dual spaces and tensor products==

The [[dual space|dual]] of a real vector space {{math|''V''}} is the space {{math|''V''*}} of all real linear maps from {{math|''V''}} to {{math|'''R'''}}. The complexification of {{math|''V''*}} can naturally be thought of as the space of all real linear maps from {{math|''V''}} to {{math|'''C'''}} (denoted {{math|Hom<sub>'''R'''</sub>(''V'','''C''')}}). That is,
<math display=block>(V^*)^{\Complex} = V^*\otimes \Complex \cong \mathrm{Hom}_{\Reals}(V,\Complex).</math>

The isomorphism is given by
<math display=block>(\varphi_1\otimes 1 + \varphi_2\otimes i) \leftrightarrow \varphi_1 + i \varphi_2</math>
where {{math|''φ''<sub>1</sub>}} and {{math|''φ''<sub>2</sub>}} are elements of {{math|''V''*}}. Complex conjugation is then given by the usual operation
<math display=block>\overline{\varphi_1 + i\varphi_2} = \varphi_1 - i \varphi_2.</math>

Given a real linear map {{math|''φ'' : ''V'' → '''C'''}} we may [[extend by linearity]] to obtain a complex linear map {{math|''φ'' : ''V''{{i sup|'''C'''}} → '''C'''}}. That is,
<math display=block>\varphi(v\otimes z) = z\varphi(v).</math>
This extension gives an isomorphism from {{math|Hom<sub>'''R'''</sub>(''V'','''C''')}} to {{math|Hom<sub>'''C'''</sub>(''V''{{i sup|'''C'''}},'''C''')}}. The latter is just the ''complex'' dual space to {{math|''V''{{i sup|'''C'''}}}}, so we have a [[natural isomorphism]]:
<math display=block>(V^*)^{\Complex} \cong (V^{\Complex})^*.</math>

More generally, given real vector spaces {{math|''V''}} and {{math|''W''}} there is a natural isomorphism
<math display=block>\mathrm{Hom}_{\Reals}(V,W)^{\Complex} \cong \mathrm{Hom}_{\Complex}(V^{\Complex},W^{\Complex}).</math>

Complexification also commutes with the operations of taking [[tensor product]]s, [[exterior power]]s and [[symmetric power]]s. For example, if {{math|''V''}} and {{math|''W''}} are real vector spaces there is a natural isomorphism
<math display=block>(V \otimes_{\Reals} W)^{\Complex} \cong V^{\Complex} \otimes_{\Complex} W^{\Complex}\,.</math>
Note the left-hand tensor product is taken over the reals while the right-hand one is taken over the complexes. The same pattern is true in general. For instance, one has
<math display=block>(\Lambda_{\Reals}^k V)^{\Complex} \cong \Lambda_{\Complex}^k (V^{\Complex}).</math>
In all cases, the isomorphisms are the “obvious” ones.