Question

In a Banach space, show that every absolutely convergent series is convergent.

Answer 1

Let $X$ be Banach space.

Let $\sum\limits_{k=1}^{\infty}a_k$ be an absolutely convergent series in $X$ , that is $\sum\limits_{k=1}^{\infty}\|a_k\|$ is a convergent series.

Denote $\sum\limits_{k=1}^na_k$ by $S_n$, and let $S_0=0$.

By the Cauchy's convergence test we have that for every $\varepsilon>0$ there is $N\in\mathbb N$ such that $\sum\limits_{k=n}^m\|a_k\|<\varepsilon$ for every $n,m>N, m\ge n$.

By the general triangle inequality we have $\|S_m-S_{n-1}\|=\bigl\|\sum\limits_{k=n}^ma_k\bigr\|\le\sum\limits_{k=n}^m\|a_k\|$, so $\{S_n\}_{n\in\mathbb N}$ is a fundamental sequence in $X$. Since $X$ is Banach space, we have that $\{S_n\}_{n\in\mathbb N}$ is a convergent sequence, that is $\sum\limits_{k=1}^{\infty}a_k$ is a convergent series.