# Incompleteness of $\ell^1$ with respect to $\sup$ norm

I'm trying to make an example that shows $\ell^1$, that is the space of complex sequences that the sum of the norms of their components is finite, is not complete with respect to $\sup$ norm.

And also a sequence of continuous linear functional on this space with $\sup$ norm that their limit is not a continuous linear one.

I've tried a lot to make such examples. Is there any hint? Thank you very much.

Let $S_k$ be the sequence $$1, 1/2, 1/3, ..., 1/k, 0, 0, 0, \ldots$$ Each $S_i$ is is in $\ell^1$, but the limit is the harmonic series, which is not.
• This example already gives an example of the second type: take $F_n(S)=\sum_{j=0}^n S(j)$. Then the limit of $F_n$ is $\sum_{j=0}^\infty S(j)$, which is not continuous in the sup norm, as the example above demonstrates. – Ian Dec 23 '15 at 17:19
• I think the answer would be better stated as $S_k$ is a Cauchy sequence in the sup norm, but there is no sequence in $l^1$ to which it converges to in the sup-norm. And then some nod towards a proof of this needs to be made. – zhw. Dec 23 '15 at 19:02