# Krein-Milman and dual spaces

I was reading a book about duality and the examples of spaces that were not dual spaces were satisfying the following property: their unit ball had no extreme point.

I was, therefore, wondering about the following facts :

• If the unit ball of a Banach space has an extreme point, is it necessarily the dual space of some other space?

• Is there a Banach space such that the unit ball has extreme points but is not their closed convex hull?

Consider the space $C[0,1]$ of continuous real functions on the unit interval with the uniform norm. The unit Ball has extreme points, namely the constant functions with values $1$ and $-1$. But the closed convex hull of the extreme points under any separated linear topology is only a line segment and not the whole unit ball. In particular, $C[0,1]$ is not a dual space, for otherwise, the unit Ball would be the weak*-closed convex hull of the two extreme points and therefore a line segment.