# Complexification of a Banach lattice

If $E$ is a real Banach lattice, then complexification of $E$ is defined as follows: $$E_c= E+iE=\{x+iy:x,y \in E\}\\(x_1+iy_1)+(x_2+iy_2)=(x_1+x_2)+i(y_1+y_2)\\(x_1+iy_1)(x_2+iy_2)=(x_1x_2-y_1y_2)+i(x_2y_1+x_1y_2)\\|x+iy|=\sup_{\theta \in [0,2\pi)}|x\cos\theta+y\sin\theta|$$

What is the motivation behind describing $|x+iy|$ as above?

For real numbers $x,y$, then there is a $\theta\in[0,2\pi)$ such that $|x+iy|=e^{i\theta}\cdot(x,y)=x\cos\theta+y\sin\theta$ as the usual dot product in the plane.
• What is the motivation begin the $\sup$ in defining $|x+iy|$? – Sahiba Arora Mar 31 '18 at 14:49
• I think it is something like to include all the rotation angle, in the complex plane, $|z|=\sup\{\text{Re}(ze^{-i\theta}): 0\leq\theta< 2\pi\}$. – user284331 Mar 31 '18 at 18:23