proof of Banach-Alaoglu 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|
|Date of creation||2013-03-22 15:10:03|
|Last modified on||2013-03-22 15:10:03|
|Last modified by||Mathprof (13753)|