I have been unable to solve the following problems from Kechris. Any hints, advice, or solutions would be appreciated.

1. Exercise 4.9: Let be separable Banach spaces. The **weak topology** on is the one generated by the functions (from into the scalar field) . Show that if is reflexive, with the weak topology is compact metrizable. Find a compatible metric.

2. Exercise 4.10:

A **topological vector space** is a vector space (over , where ), equipped with a topology in which addition and scalar multiplication are continuous. So Banach spaces and their duals with the weak*-topology are topological vector spaces. A point in a convex set is **extreme** (in ) if , with , , implies . Denote by the **extreme boundary** of , i.e., the set of extreme points of .

Show that if is a compact metrizable (in the relative topology) convex subset of a topological vector space, the the set is in .