(1) Let be an orthonormal base of the Hilbert space X. We define and :

,

Prove that isn't closed.

I've been trying to find a convergent sequence in , with a limit outside , but no success. (Trying to prove that its complement is open would be much harder, I think.)

(2) Let be an orthonormal base of the Hilbert space X with the inner product

We define the operator as follows:

.

Is bounded, normal?

Is it true that , where S is the unilateral shift?

This one is just too messy. I can post what I've been trying to do, but didn't get anywhere and I think it wouldn't do any good.

Thank you once again for all your help.