# real analysis – Does the linear span of norming extreme points contain all extreme points?

Let $$X$$ be a Banach space, $$X^*$$ be its continuous dual space, and $$B(X^*)$$ be its closed unit ball, i.e., $$B(X^*) = {x^*in X^*: |x^*|leq 1}$$.

We say that $$x^*$$ is an extreme point of $$B(X^*)$$ if $$x^*in B(X^*)$$ and for any $$x^* = frac{1}{2}(x_1^* + x_2^*)$$ for some $$x_1^*,x_2^*in B(X)$$, we have $$x^* =x_1^*=x_2^*$$.

We say that $$x^*$$ is a norming extreme point of $$B(X)$$ if $$x^*$$ is an extreme point of $$B(X)$$ that satisfies $$x^*(x) = |x|$$ for some $$xin Xsetminus {0}$$.

Question: If $$x^*$$ is an extreme point of $$B(X)$$, then is it true that
$$x^* = a_1x_1^* + cdots +a_n x_n^*$$
for some $$ngeq 1$$, $$a_1,…,a_nin mathbb{R}$$ and $$x_1^*,…,x_n^*$$ are norming extreme points of $$B(X^*)$$?