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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.

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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.

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  • $\begingroup$ oh!that's true!i considered it much harder!but what about the second? $\endgroup$ – user115608 Dec 23 '15 at 17:17
  • $\begingroup$ 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. $\endgroup$ – Ian Dec 23 '15 at 17:19
  • $\begingroup$ Thanks, Ian. I completely forgot about the second part as I was writing. :( $\endgroup$ – John Hughes Dec 23 '15 at 17:38
  • $\begingroup$ 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. $\endgroup$ – zhw. Dec 23 '15 at 19:02
  • $\begingroup$ I figured I'd given the OP the starting point for all that, which seems in practice to have sufficed. $\endgroup$ – John Hughes Dec 24 '15 at 18:30

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