I'm reading a solution to the following problem: Prove that no two dimensional subspace of $c_0$ is smooth.
Proof: Suppose this is not true. Then some quotient $Q$ of $\ell^1$ would have uncountably many extreme points, since the dual of a two-dimensional smooth space is strictly convex. Every point of the sphere of $Q$ that is identified with the restriction to the two dimensional subspace in question extends to an extremal point of the sphere in $\ell^1$ by the Krein Milman theorem, considering the face of all the extensions. Thus, there are uncountably many such extreme points of the ball of $\ell^1$, which is a contradiction because the extreme points of the ball of $\ell^1$ are precisely $\pm e_i$.
I don't quite understand what the author means by, "every point in the sphere of $Q$... considering the face of all the extensions". I think what's he's using is the partial converse to Krein Milman, which says that if $C$ is compact, then every extreme point in $\overline{co}(C)$ is in $C$, along with the fact that the ball of $\ell^1$ is weak star compact. Could someone explain?