Let $X$ be a linear space over $\mathbf{C}$. If $Z_r$ is a real hyperspace in $X$, then $Z_r\cap iZ_r$ is a complex hyperspace in $X$.
I know, we have to use the fact: $Ref$ determines $f$ as follows: $$f(x)=Ref(x)-iRef(ix).$$ But I want rigorous proof, I have tried it but the imaginary part is creating some problem. Thank you, beforehand.
