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 net
in , 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.
Title | proof of Banach-Alaoglu theorem |
---|---|
Canonical name | ProofOfBanachAlaogluTheorem |
Date of creation | 2013-03-22 15:10:03 |
Last modified on | 2013-03-22 15:10:03 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 12 |
Author | Mathprof (13753) |
Entry type | Proof |
Classification | msc 46B10 |