Let X be a Banach space and $f_i \in X^*, i \in \mathbb{N}$ such that $$ \sum_{i=1}^\infty|| f_i(x)|| < \infty, \forall x \in X. \hspace{2cm}(I) $$
Calculate the norm of $T: X \to l^1(\mathbb{N})$ defined as $T(x)= (f_1(x),f_2(x),\dots)$.
I managed to prove that $T$ is continuous via the Closed Graph Theorem, but I could not use this technique to find its norm.
My first guess was $\sum_{i=1}^\infty || f_i||$, but this is not necessarily finite, even with hypothesis, as I could take $X = l^1(\mathbb{N})$ and taking the $\{f_i\}_{i\in \mathbb{N}}$ to be the projections in its coordinates.
I can't really see what answer you're expected to give here. The norm of $T$ certainly isn't determined by just the norms of the $f_i$. For instance, if you take $f_1=f_2=\dots=f_n$ and $f_i=0$ for $i>n$, then $\|T\|=n\|f_1\|$. On the other hand, your example with $X=l^1(\mathbb{N})$ shows that it's possible to have $\|f_i\|=1$ for all $i$ but $\|T\|=1$. By similar examples along these lines, it should be possible to show that $\|T\|$ can be any finite value between $\sup \|f_i\|$ and $\sum \|f_i\|$.