Alaoglu's Extension of Helly's Theorem

For $X$ a normed linear space with closed unit ball $B$, suppose that the function $f: B \rightarrow [-1,1]$ has the property that whenever $u, v, u+v$, and $\lambda u$ belong to $B, f(u+v) = f(u)+ f(v)$ and $f(\lambda u) = \lambda f(u)$. Show that $f$ is the restriction to $B$ of a linear functional on all of $X$ which belongs to the closed unit ball of $X^{*}$.

Let $X$ be a normed linear space and $K$ be a bounded convex weak-* closed subset of $X^{*}$. Show that $K$ possesses an extreme point.

Show that any non-empty weakly open set in an infinite dimensional normed linear space is unbounded with respect to the norm.

Use the Baire Category Theorem and the preceding problem to show that the weak topology on an infinite dimensional Banach space is not metrizable by a complete metric.

Is every Banach space isomorphic to the dual of a Banach space?