Let $X$ be a Banach space. And let $X^\* $ be the dual space of $X$. Let $E\_X$ and $E\_{X^\*}$ denote the extreme points of the unit ball of $X$ and $X^\*$. Let $x\in X$ and $|f(x)|=1$ for every $f\in E\_{X^\*}.$ Does that imply $x\in E\_X?$


Can anyone suggest a text to study theory of extreme points of a convex set in a Banach space(for beginners)?



Regarding extreme point in a Banach space
Suppose that $x$ satisfies the condition and is not extreme in the unit ball of $X$. It means that we can write $x = \frac{y+z}{2}$ for some $y,z \in B\_{X}$ different from $x$. Since $|\frac{1}{2}f(y)+\frac{1}{2}f(z)|=1$, we get that $f(y)=f(z)$, so $f(y-z)=0$ and $y-z\neq0$. It holds for any $f$ in $E\_{X^{\ast}}$, so it also holds for arbitrary $f\in B\_{X^{\ast}}$ by the Krein-Milman theorem. This is a contradiction, since the dual separates points.


I seem to have avoided the closure issue noted by Dirk Werner in the comment section. Is this correct?


