# Extreme unit linear functional not norming a vector

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 $E$?

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.
• As an aside, is there then an analogue of the Bishop-Phelps theorem restricted to extreme norm one linear functionals? All I can prove is weak$^*$ density (which is enough for my needs). – Itaï BEN YAACOV Oct 16 '14 at 18:41