Complexification - Dual Spaces and Tensor Products

Dual Spaces and Tensor Products

The dual of a real vector space V is the space V* of all real linear maps from V to R. The complexification of V* can naturally be thought of as the space of all real linear maps from V to C (denoted Hom_R(V,C)). That is,

The isomorphism is given by

where φ₁ and φ₂ are elements of V*. Complex conjugation is then given by the usual operation

Given a real linear map φ : V → C we may extend by linearity to obtain a complex linear map φ : VC → C. That is,

This extension gives an isomorphism from Hom_R(V,C)) to Hom_C(VC,C). The latter is just the complex dual space to VC, so we have a natural isomorphism:

More generally, given real vector spaces V and W there is a natural isomorphism

Complexification also commutes with the operations of taking tensor products, exterior powers and symmetric powers. For example, if V and W are real vector spaces there is a natural isomorphism

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

In all cases, the isomorphisms are the “obvious” ones.