proof of Banach-Alaoglu theorem
For any , let and . Since is a compact subset of , is compact in product topology by Tychonoff theorem.
We prove the theorem by finding a homeomorphism that maps the closed unit ball of onto a closed subset of . Define by and by , so that. Obviously, is one-to-one, and a net in converges
to in weak-* topology
of iff converges to in product topology, therefore is continuous
and so is its inverse
.
It remains to show that is closed. If is a netin , converging to a point , we can define a function by . As for all by definition of weak-* convergence, one can easily see that is a linear functional in and that . This shows that is actually in and finishes the proof.