A question about extreme points If the extreme points of the unit ball of $C[0, 1]$ are $\pm{1}$, where $C[0, 1]$ is the Banach space of all continuous real-valued functions on the unit interval, then what would the extreme points of the unit ball be if we considered all continuous complex-valued functions on the unit interval? Would the extreme points in that case include all complex numbers $z$ such that $|z|= 1$?

The extreme points would be all continuous functions $f$ with $|f(t)| = 1$ for all $t \in [0,1]$.  
