# Proving vector spaces are banach spaces

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?

• Try to prove the completeness property in your space. – Mhenni Benghorbal Dec 11 '13 at 6:12
• I can prove it for the space of convergent series, but I'm not sure if it holds for absolute convergent series as well. – Bob Hsu Dec 11 '13 at 6:14
• You should give it a try. – Mhenni Benghorbal Dec 11 '13 at 6:42
• 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 – Bob Hsu Dec 11 '13 at 6:51

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.