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Is the completion of $\{x=(x_n)|x_n\in \mathbb R \text{ and  for a given } x,\text{ only finitely many } x_n\neq0\}$ equipped with the norm $\|x\|:= |x_1|+|x_2|+...$ simply the set of all real sequences?

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    I think you want to say "only finitely many $x_n\ne0$"; otherwise, $\Vert x\Vert$ could be infinite. If this is what you meant, then the completion will be the set of absolutely summable sequences (that is, $\ell_1$).2011-12-09
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    @DavidMitra: Thanks! I misread my notes, you are right.2011-12-09

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