1) $c_{00} \owns (x_n) \mapsto \sum_{n=0}^{\infty} x_n \in \mathbb{K}$
where $c_{00}$ is a space of sequences that are eventually equal to $0$ with sup norm
2) $\ell^\infty \owns (x_n) \mapsto \left(\frac{1}{n+1} x_n\right)\in\ell^2$
My attempt
1) The first one is quite obvious. There is no universal constant $C$ which satisfies
$\left|\sum_{n=0}^{\infty} x_n\right| \le C \cdot\sup_n |x_n|$ since we can always replace the first zero term with a non-zero one. So 1) is not continuous. Right?
2) It's somehow similar to 1), but this time I'm not so sure. Can anyone give me any clue?
1) For every $n$, $x_n=(\underbrace{1,\ldots,1}_{n\mathrm{\ times}},0,\ldots)\in c_{00}$, $\Vert x_n\Vert=1$ but $x_n$ maps to $n$, so this function is not continuous
2)Let $\phi:\ell^\infty \ni (x_n)\mapsto (\frac{1}{n+1}x_n)\in \ell^2$. Since $$\Vert \phi(x_n)\Vert_2=\left(\sum\left(\frac{1}{n+1}\right)^2|x_n|^2\right)^{1/2}\leq\Vert(x_n)\Vert_{\infty}\sum\frac{1}{(n+1)^2}(<\infty),$$ $\phi$ is well-defined, continuous, and and $\Vert\phi\Vert\leq\sum\frac{1}{(n+1)^2}$. Also, $\Vert\phi(1,1,\ldots)\Vert=\sum\frac{1}{(n+1)^2}$, so $\Vert\phi\Vert=\sum\frac{1}{(n+1)^2}$.