Normed linear space and linear functional Let $X$ be the normed linear spaceof sequences of reals that have only finitely many non-zero terms. Given $x = \{x_n\} \in X$, define
$$f(x) = \displaystyle \sum_{n=1}^{\infty} x_n$$
I think that it is quite easy to prove that $f$ is linear (please correct me if you think otherwise). My question is whether $f$ is continuous if we give $X$ the topology induced by the $l^{\infty}$ norm.
Thank you very much in advance!

No. Consider $B(0,2\varepsilon) \subset X$ and let 
$$y_n=\sum_{k=1}^n \varepsilon e_k$$
where $e_k$ is the usual basis for $X$. Then $y_n \in B(0,2\varepsilon)$ and $f(y_n)=\varepsilon n$ in particular the image of any neighborhood around $0$ is unbounded. It follows that we can't find an open set containing $0$ in $f^{-1}((-\varepsilon,\varepsilon))$ whose image is contained in $(-\varepsilon,\varepsilon)$. 
