(1) Let $\displaystyle \{e_n, n \in \mathbb{N}\}$ be an orthonormal base of the Hilbert space X. We define $\displaystyle Y_1$ and $\displaystyle Y_2$:

$\displaystyle Y_1=\overline{L\{e_{2n}, n \in \mathbb{N}}\}$, $\displaystyle Y_2=\overline{L\{e'_n=e_{2n}cos1/n + e_{2n+1}sin 1/n, n \in \mathbb{N} \}}$

Prove that $\displaystyle Y_1 + Y_2$ isn't closed.

I've been trying to find a convergent sequence in $\displaystyle Y_1 + Y_2$, with a limit outside $\displaystyle Y_1 + Y_2$, but no success. (Trying to prove that its complement is open would be much harder, I think.)

(2) Let $\displaystyle \{e_n, n \in \mathbb{N}\}$ be an orthonormal base of the Hilbert space X with the inner product $\displaystyle <. | .>$

We define the operator $\displaystyle T_n$ as follows:

$\displaystyle T_n x= <x|e_n>e_1 + <x|e_1>e_2 + <x|e_2>e_3 + \ldots +$

$\displaystyle <x|e_{n-1}>e_n + <x|e_{n+1}>e_{n+1} + <x|e_{n+2}>e_{n+2} + \ldots, n \in \mathbb{N}$.

Is $\displaystyle T_n$ bounded, normal?

Is it true that $\displaystyle \forall x \in X$ $\displaystyle T_nx \rightarrow Sx$, 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.