Exercise 2 [35 Marks] For a sequence $u=\left(u_{k} ight)_{k \in Z} \in \mathbb{R}^{Z}$, we define $$ Nu\|_{2}=\left(\sum_{k \in Z}\left|u_{k} ight|^{2} ight)^{1 / 2] $$ And we let $\ell^{2}=\left\{u=\left(u_{k} ight) {k \in Z} \in \mathbb{R}^{Z}:\]\]_{2} $ and $\left(u_{k} ight)_{k=1}^{\infty} \subset H$ is a sequence of unit vectors. We define $T: \mathcal{H} ightarrow \mathbb{C}$ by $$ Tu=\sum_{k=1}^{\infty} \frac{1}{k^{2}}\left\langle u, u_{k} ight angle $$ 2-2-1.. Show that $T$ is well defined and that it is a bounded linear functional. [15] 2-2-2. Compute the operator norm $\T\|$. [10] God bless you !!! SP.SD. 312