Definition: Let $X$ be a Banach space and $X^*$ be its continuous dual of $X,$ that is, $X^*$ contains all bounded linear functionals on $X.$
Denote
$$B_{X^*} = \{x^*\in X^*: \|x^*\|_{X^*}\leq 1\}.$$
We say that $x^*\in B_{X^*}$ is an **extreme point of** $B_{X^*}$ if whenever
$$x^* = \frac{1}{2}(y_1^*+y_2^*)$$
for some $y_1^*,y_2^*$ with $\|y_i^*\|_{X^*} \leq 1$ for all $i=1,2,$ we have $x^* = y_1^*=y_2^*.$

Question: Given a Banach space $X.$ Is it true that there exists an extreme point $x^*$ of $B_{X^*}$ such that $$x^*(x) = 1$$ for some $x\in X$ with $\|x\|\leq 1?$

In other words, is it true that every Banach space has at least one extreme point that is normed by some point?

extreme pointnot of $B_X$ but of $B_{X^*}$. And then you ask a question aboutextreme functionalnotextreme point. I think you should re-write the question. $\endgroup$ – Gerald Edgar Oct 24 '18 at 13:35