I want to show the weak star closed convex hull of a finite set of points is contained in the linear span of those points.
It's enough to show that any finite dimensional subspace $V$ of a Banach space $Z$ is weak star closed in $Z$. Since $V$ is finite dimensional, it is a closed subspace of $Z$ in the norm topology. How can I show that it is also weak star closed? Also, it this result true for arbitrary subspaces?