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I would like to find

  • a vector space $E$
  • a norm on $E$
  • a sequence $(u_n)$ which converges for this norm but such that its components doesn't converge to the component of the limit.

First $E$ should be infinite dimensional.

Second I am tying to look at $E=\mathbb{R}[X]$.

Take $u_n=(\underbrace{0,\ldots,0}_{\text{n zeros}},1/n,1/(n+1),\ldots,1/(n+p),\ldots)$.

or

$u_n=(1/1,1/2,\ldots,1/n,0,\ldots,0,\ldots)$.

We have $||u_n||_1=\sum_{k=1}^n1/k$

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  • $\begingroup$ The question does not make much sense. Please specify the space of sequences you are considering and specify the norm you want to use. $\endgroup$ – Kavi Rama Murthy Feb 11 at 10:14
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    $\begingroup$ This can be an answer, if I understand the question properly. $\endgroup$ – Giuseppe Negro Feb 11 at 10:18

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