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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.