Definition: Let $X$ be a Banach space. We say that a bounded linear functional $x^*:X\to \mathbb{R}$ is an extreme point of $B_X = \{x\in X:\|x\|_X\le 1\}$ 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 bounded linear extreme functional $x^*:X\to\mathbb{R}$ 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 functional that is normed by some point?

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?