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Show that a Normed vector space $\mathbb{X}$ is Banach space iff for every sequence $(x_n)$ in $\mathbb{X}$,such that $\Sigma \|x_n\| <\infty$, the series $\Sigma x_n$ converges in $\mathbb{X}$.

I know that a complete (where every Cauchy sequence converges) normed space is a Banach space. So if I take a Cauchy sequence $(x_n)$ in $\mathbb{X}$ then I need to prove that the sum of the sequence converges.

Thanks in advance.


marked as duplicate by Rhys Steele, Theoretical Economist, hardmath, Michael Greinecker Aug 22 '18 at 17:40

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    $\begingroup$ Have you seen the proof of dominated convergence where one picks a subsequence that converges very fast? You might want to check out that technique, because it might be helpful. $\endgroup$ – 4-ier Aug 22 '18 at 6:26
  • $\begingroup$ @4-ier Could you explain how that would help in my case? $\endgroup$ – antony james Aug 22 '18 at 6:55
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    $\begingroup$ You seem to be partly confused. To prove the result, you need to show. 1. If $X$ is Banach then every absolutely convergent series converges. (Just need to show that the partial sums are Cauchy). and 2. If every absolutely convergence series converges, then $X$ is Banach. (This step is more involved: Take a Cauchy sequence $x_n$. And find a subsequence $x_{n_k}$ such that $\sum_k\|x_{n_k}-x_{n_{k+1}}\|$ converges, then conclude.) This was just a rough sketch and you would need to fill in the details. $\endgroup$ – 4-ier Aug 22 '18 at 7:00