How do I find the norm of the following operator i.e. how to find $\lVert T_z\rVert$ and $\lVert l\rVert$?
1) Let $z\in \ell^\infty$ and $T_z\colon \ell^p\to\ell^p$ with $(T_zx)(n)=z(n)\cdot x(n).$ What my thoughts were to use Banach-Steinhaus theorem but it seems straight forward and I don't know if I am right.
$\lVert T_z\rVert _p \leqslant\lVert z\lVert \cdot n\cdot\lVert x\rVert_p n=n^2\lVert x\rVert _p$ so if I choose $x=1$ then I get $\lVert T_z\rVert =n^2$.
2) Let $0\leqslant t_1\leqslant\cdots\leqslant t_n=1$ and $\alpha_1,\dots,\alpha_n \in K$ , $l\colon C([0,1])\to K$ with $l(x)=\sum_{i=1}^n \alpha_i x(t_i)$.
How to I find operator norm in this case as well? I am quite sure I am not right. I would be glad if I could get some help. Definitely some hints would be great! Thanks in advance.