finite rank approximation on separable Hilbert spaces
It is clear that the have finite rank and that we have for all , .
Let be the unit ball in . We have that pointwise. Since the are contractive they are equicontinuous, hence converges uniformly to on compact sets, and in particular on , which is compact by assumption. Therefore uniformly on , hence . Since is bounded and of finite rank the first direction follows.
: Now let be a sequence of bounded operators of finite rank with . We have to show that is relatively compact in . This is equivalent to being totally bounded in . So we are left to show that for all there is an -net so that:
So choose and fixed so that:
Hence (by the triangle inequality):
and we are done. ∎
|Title||finite rank approximation on separable Hilbert spaces|
|Date of creation||2013-03-22 18:23:18|
|Last modified on||2013-03-22 18:23:18|
|Last modified by||karstenb (16623)|