Maximum attained on extreme points of the ball in $\mathcal{L}(^2\ell^3_\infty)$

Question

Let $B$ be the closed unit ball in $\mathcal{L}(^2\ell^3_\infty)$ (over $\Bbb R$), $\text{ext}B$ be the set of its extreme points and $f\in\mathcal{L}(^2\ell^3_\infty)^*$. How can I show that $$\|f\|=\sup_{x\in \text{ext}B}|f(x)|.$$

It seems that we can't use the Krein-Milman Theorem because $B$ is not compact. Do you know another approach?

Answer 1

A normed space is reflexive iff $B$ is weakly compact. Apply now Bauer's Maximum Theorem with $|f|,$ $B,$ and considering the weak topology.

I haven't shown that your space is reflexive, so...