Real structure

In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces. The prototype of such a structure is the field of complex numbers itself, considered as a complex vector space over itself and with the conjugation map ${\displaystyle \sigma :{\mathbb {C} }\to {\mathbb {C} }\,}$, with ${\displaystyle \sigma (z)={\bar {z}}}$, giving the "canonical" real structure on ${\displaystyle {\mathbb {C} }\,}$, that is ${\displaystyle {\mathbb {C} }={\mathbb {R} }\oplus i{\mathbb {R} }\,}$.

The conjugation map is antilinear: ${\displaystyle \sigma (\lambda z)={\bar {\lambda }}\sigma (z)\,}$ and ${\displaystyle \sigma (z_{1}+z_{2})=\sigma (z_{1})+\sigma (z_{2})\,}$.