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!

Observe that for $(x^k)_k=((x_n^k)_n)_k \subset X$, where $x_n^k = 1$ for $n \leq k$ and $x_n^k = 0$ otherwise, we have $\lVert x^k \rVert_{\ell^\infty} = 1$ and
$$
f(x^k) = k.
$$
Hence $\lim_{k \to \infty} f(x^k) = \infty$ which proves that $f$ cannot be continuous.
