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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 $E$?

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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.

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  • $\begingroup$ 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). $\endgroup$
    – Itaï BEN YAACOV
    Commented Oct 16, 2014 at 18:41
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    $\begingroup$ Yes, if the dual of the space is separable or, more generally, if every separable subspace has separable dual ("Asplund space"). Then the dual ball is the norm closed convex hull of its extreme points. No, in general (consider $C[0,1]). $\endgroup$
    – Bill Johnson
    Commented Oct 16, 2014 at 19:01
  • $\begingroup$ Thanks but that is not what I meant - given an extreme point of the dual ball, can it be expressed as a norm limit of extreme points which norm something? No convex hull taken. $\endgroup$
    – Itaï BEN YAACOV
    Commented Oct 17, 2014 at 6:09
  • $\begingroup$ @BillJohnson Do have any idea about: If $X$ is non-reflexive such that $X^*$ is not strictly convex then there always exists $f\in \operatorname{ext}(B_{X^*})$ such that $f$ is not norm attaining? $\endgroup$
    – Tuh
    Commented Nov 27 at 11:08

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