Hello all, I am having trouble with this problem and was wondering if anyone can help. I will post my partial proof and put the questions in bold, thanks very much!
Let H be a Hilbert space and let V,W be linear subspaces such that . Prove that is closed are both closed.
( ) Let both be closed. Let be a convergent sequence in with . We have where is a sequence in and is a sequence in . Now if and both converge then and , where , but what if and are divergent? Is it not possible they diverge but both somehow converge to when you add them together?
( ) Let be closed. Let be a sequence in . Then , so . How do I show that ?
I would really appreciate any help with this, thank you