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