Due to several discussions on the main site, I am asking this question concerning the best practices for tagging questions on MathOverflow. Unfortunately, the relevant documentation page https://mathoverflow.net/help/tagging does not answer the following questions (except partially questions numb...

Let $X$ be a Banach space. By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball. I am interested in its converse. More precisely, Question: Let $X$ be a Banach space. If the closed uni...

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...