I've been given the following question to do as an exercise and i'm really not sure on how to do it, or even really start it. If any hints or help can be given it would be really appreciated.
Let $$ l_\infty=\{\{x_n\}_{n\in\mathbb{N}}:\exists C>0:|x_n|\le C, n\in \mathbb{N}\} $$ With the norm $$ ||x||_\infty =sup\{|x_n|:n \in \mathbb{N}\} $$ Consider for some $y=\{y_n\}_{n \in \mathbb{N}} \in l_\infty$ the operator $$T_y:l_\infty \to l_\infty, (T_yx)_n=y_nx_n$$ a) Show that indeed $T_yx\in l_\infty$ for all $x \in l_\infty$
b)Compute $||T_y||$
a) As $T_y x = (x_1 y_1,\dots,x_n y_n,\dots),$ it is not hard to observe that $\|T_y x\|_{\ell_\infty} \leq \|y\|_{\ell_\infty} \|x\|_{\ell_\infty} .$ Thus $$\|T_y\| \equiv \|T_y\|_{\mathcal{L} (\ell_\infty\,,\ell_\infty)} \leq \|y\|_{\ell_\infty} \,.$$
b) In fact, $\|T_y\| = \|y\|_{\ell_\infty}.$
For any $\varepsilon >0,$ assume that $|y_{N}| > \|y\|_{\ell_\infty} -\varepsilon$ for some large $N \in \mathbb{N}.$ Take $ x_0 = (\delta_{1N}, \dots, \delta_{nN},..)$ with $\|x_0\|_{\ell_\infty} =1,$ and then we have $$\|T_y\| \geq \|T_y x_0\| \geq |y_N| > \|y\|_{\ell_\infty} -\varepsilon.$$ Therefore combining a), we obtain $\|T_y\| = \|y\|_{\ell_\infty}\,.$