Realification of a complex vector space

Core idea

Let $V$ be a complex vector space. Its realification $V_{\mathbb R}$ is the same additive group, regarded as a real vector space by restricting scalar multiplication along $\mathbb R\hookrightarrow\mathbb C$ . If $\dim_{\mathbb C}V=n<\infty$ , then

\dim_{\mathbb R}V_{\mathbb R}=2n.

Complex structure retained as an operator

Multiplication by $i$ becomes a real-linear endomorphism

J:V_{\mathbb R}\to V_{\mathbb R},\qquad J^2=-\operatorname{id}.

Conversely, a real vector space equipped with such an operator $J$ becomes a complex vector space by defining $(a+ib)v=av+bJv$ .

Functoriality

Every complex-linear map $f:V\to W$ is real-linear after realification and commutes with $J$ . Not every real-linear map $V_{\mathbb R}\to W_{\mathbb R}$ arises this way: the additional condition is $fJ_V=J_Wf$ .

References

Steven Roman, Advanced Linear Algebra, 3rd ed., Springer, 2008. DOI record. Relevant: restriction of scalars and complex structures.