I've been working on closed range theorem. There are a lot of materials on general Banach spaces, but not much on Hilbert spaces, so I was wondering if I could get some help.
I'm trying to prove the following claim:
If a bounded linear map $T:X\to Y$ between Hilbert spaces $X$ and $Y$ has closed range if and only if there exists a constant $C>0$ so that $\|f\| \leq C\|T^*f\|$
This statement seems like the statement is a usual closed range theorem, but a bit different, especially with adjoint of the operator. Can someone help me proving this claim? Thanks!
Best Answer
The statement is wrong: $T=0$ has closed range but the inequality is obviously false.