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Let $V$ be the vector space of absolutely convergent series of real numbers under pointwise operations and with norm $\|\{a_n\}\| = \sum_n |a_n|$. Is $V$ a Banach space?

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  • $\begingroup$ Try to prove the completeness property in your space. $\endgroup$ – Mhenni Benghorbal Dec 11 '13 at 6:12
  • $\begingroup$ I can prove it for the space of convergent series, but I'm not sure if it holds for absolute convergent series as well. $\endgroup$ – Bob Hsu Dec 11 '13 at 6:14
  • $\begingroup$ You should give it a try. $\endgroup$ – Mhenni Benghorbal Dec 11 '13 at 6:42
  • $\begingroup$ Any hints? I've tried for a while already, knowing all I needed to do was to prove that to show it's a banach space $\endgroup$ – Bob Hsu Dec 11 '13 at 6:51
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Hint: Complete your proof in the following steps:

Step 1: Let $\{\{a_n^1\}, \{a_n^2\}, \cdots\}$ be a Cauchy sequence in $V$. Use this to show that the sequence $\{a_p^m\}_{m\in\mathbb{N}}$ (i.e., the sequence whose $m$-th term is the $p$-th term of $\{a_n^m\}_{n\in\mathbb{N}}$) is convergent for all $p\in\mathbb{N}$. (hint: use the completeness of $\mathbb{R}$).

Step 2: Let $a_p^m\to a_p^*$ (by your conclusion from the last step). Show that $\{a^*_p\}_{p\in\mathbb{N}}\in V$.

Step 3: Show that $\{a_p^*\}_{p\in\mathbb{N}}$ is rightly the limit of $\{\{a_n^1\}, \{a_n^2\}, \cdots\}$. Conclude.

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