Let $H$ be an infinite dimensional separable Hilbert space with orthonormal basis $(e_n)_{n\geq1}$ and let $D$ be the norm closure of $co\{\frac{1}{N}\sum \limits_{n=1}^{N^2}e_N : N ≥ 1\}$.
I want to show that $D$ is weakly compact, and that $0 ∈ D$ as well as to find the extreme points in $D$.
Guess for some of the extreme points in $D$: 0, as well as each $\frac{1}{N}\sum \limits_{n=1}^{N^2}e_N, N ≥ 1$,