proof of open mapping theorem
Let , be the open unit balls in , respectively. Then , so, since is surjective, . By the Baire category theorem, is not the union of countably many nowhere dense sets, so there is some and some open set such that is contained in the closure of .
Let , and pick so that for all with . Then and are limit points of , so there are sequences and in with and . Letting , we have and . So for any there is a sequence in with . Then by the linearity of , we have that for any and any , there is an with:
Now let and . Then there is some with and . Define a sequence inductively as follows. Assume:
Then by (1) we can pick so that:
and , so (2) is satisfied for .
|Title||proof of open mapping theorem|
|Date of creation||2013-03-22 16:23:31|
|Last modified on||2013-03-22 16:23:31|
|Last modified by||Statusx (15142)|