If $E$ is a non-reflexive Banach space then there exists a linear functional $\lambda \in E^\*$ of norm one such that $\lambda v < 1$ for all $v \in E$ of norm one. However, in the only non-reflexive examples that I somewhat understand ($\ell\_1$, $c\_0$) this does not happen when $\lambda$ is an extreme point of the closed unit ball of $E^\*$.


Question: does there exist a Banach space $E$ and $\lambda \in E^\*$ such that $\lambda$ is an extreme point of the unit ball and yet does not norm any (non-zero) vector of $E$?



Extreme unit linear functional not norming a vector
Would you like a solution that does not involve any example that is new for you?? By James' theorem, which you mentioned, it would be enough to have a space $E$ that is not reflexive but every linear functional of norm one is an extreme point of the unit ball of $E^\*$. In other words, you want the dual norm on $E^\*$ to be strictly convex. But M. M. Day, Strict convexity and smoothness of normed spaces, Trans. Amer. Math. Soc. 78
(1955) 516–528, proved that every separable space $E$ has an equivalent norm whose dual norm is strictly convex. So you only need to know one example--a separable non reflexive space.


