Finding the norm of the operators 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.

*

*As for each $n$, $|z(n)x(n)|^p\leqslant \lVert a\rVert_{\infty}|x(n)|^p$, then we certainly have $\lVert T_z\rVert\geqslant \lVert a\rVert_{\infty}$. To get the other inequality, fix $\delta$ and pick $k$ such that $|a(k)|\geqslant \lVert a\rVert_{\infty}-\delta$ (the case $a=0$ is obvious). 

*We assume $t_j$ distinct. Let $f_j$ a continuous map such that $f_j(t_j)=e^{i\theta_j}$, where $e^{i\theta_j}\alpha_j=|\alpha_j|$ and $f_j(t_k)=0$ if $k\neq j$. We can choose the $f_j$'s such that $\lVert \sum_{j=1}^nf_j\rVert=1$. 
