Complex inner product/Real part/Remark

If we consider a complex vector space ${\displaystyle {}V}$, endowed with an inner product ${\displaystyle {}\left\langle -,-\right\rangle }$, as a real vector space, then the real part

${\displaystyle \operatorname {Re} \,{\left(\left\langle v,w\right\rangle \right)}}$

is a real inner product, see exercise. Because of

${\displaystyle {}{\begin{aligned}\left\langle v,w\right\rangle &=\operatorname {Re} \,{\left(\left\langle v,w\right\rangle \right)}+{\mathrm {i} }\operatorname {Im} \,{\left(\left\langle v,w\right\rangle \right)}\\&=\operatorname {Re} \,{\left(\left\langle v,w\right\rangle \right)}-{\mathrm {i} }\operatorname {Re} \,{\left({\mathrm {i} }\left\langle v,w\right\rangle \right)}\\&=\operatorname {Re} \,{\left(\left\langle v,w\right\rangle \right)}-{\mathrm {i} }\operatorname {Re} \,{\left(\left\langle iv,w\right\rangle \right)},\end{aligned}}}$

we can reconstruct from the real part the original inner product.