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.
Proof
(
) 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