In this section I only care about the weak and the weak*
topologies. In particular, the important result here is Alaoglu's
Theorem.

The book's proof of Alaoglu uses nets, so I'm posting an
alternate proof in the notes directory.

You may omit:

topological vector space
locally convex
Theorem 5.14
Propositions 5.15-17
Cauchy
complete
Frechet space
strong and weak operator topologies in general (we only need the
 case where Y is the scalar field)

Exercises 43, 44, 45, 46, 47a, 48d, 49d, 52, 53