Conjugations and real structures on a complex vector space

Let $W$ be a complex vector space. A conjugation (also called a real structure) on $W$ is a map $\sigma :W\to W$ such that for all $w,w^{\mathrm{\prime }}\in W$ and $z\in \mathbb{C}$:

1. $\sigma (w+w^{\mathrm{\prime }})=\sigma (w)+\sigma (w^{\mathrm{\prime }})$;
2. $\sigma (zw)=\bar{z}\sigma (w)$, where $\bar{z}$ is complex conjugation;
3. $\sigma (\sigma (w))=w$.

Thus a conjugation is additive, conjugate-linear (anti-linear) in the scalar action, and an involution. When $V$ is a real vector space, the canonical conjugation on the complexification $V_{\mathbb{C}}$ of Complexification as $\mathbb{C}{\otimes }_{\mathbb{R}}V$ with its canonical real-linear embedding is

${\sigma }_{\mathrm{can}}(z\otimes v):=\bar{z}\otimes v;$

its well-definedness on the tensor product follows from Universal property of the tensor product for balanced maps into abelian groups, applied to the $\mathbb{R}$-bilinear map $(z,v)\mapsto \bar{z}\otimes v$.

On a nonzero complex vector space, a conjugation is not complex-linear: if it were, then for every $w$ one would have both $\sigma (iw)=-i\sigma (w)$ and $\sigma (iw)=i\sigma (w)$, forcing $w=0$ because $\sigma$ is an involution. On the zero space the unique conjugation is also complex-linear. Every conjugation is $\mathbb{R}$-linear, because $\bar{r}=r$ for every $r\in \mathbb{R}$.