Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach/Necessary Condition
Theorem
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $X$.
Suppose $X$ is a Banach space.
Then $\ds \sum_{n \mathop = 1}^\infty a_n$ is convergent.
Proof
That $\ds \sum_{n \mathop = 1}^\infty a_n$ is absolutely convergent means that $\ds \sum_{n \mathop = 1}^\infty \norm {a_n}$ converges in $\R$.
Hence by Convergent Sequence in Normed Vector Space is Cauchy Sequence:
- the sequence of partial sums is a Cauchy sequence.
Now let $\epsilon > 0$.
Let $N \in \N$ such that for all $m, n \in \N$, $m \ge n \ge N$ implies that:
- $\ds \sum_{k \mathop = n + 1}^m \norm {a_k} = \size {\sum_{k \mathop = 1}^m \norm {a_k} - \sum_{k \mathop = 1}^n \norm {a_k} } < \epsilon$
This $N$ exists because the sequence is Cauchy.
Now observe that, for $m \ge n \ge N$, one also has:
\(\ds \norm {\sum_{k \mathop = 1}^m a_k - \sum_{k \mathop = 1}^n a_k}\) | \(=\) | \(\ds \norm {\sum_{k \mathop = n + 1}^m a_k}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{k \mathop = n + 1}^m \norm {a_k}\) | Triangle inequality for $\norm {\, \cdot \,}$ | |||||||||||
\(\ds \) | \(<\) | \(\ds \epsilon\) |
It follows that the sequence of partial sums of $\ds \sum_{n \mathop = 1}^\infty a_n$ is Cauchy.
As $X$ is a Banach space, this implies that $\ds \sum_{n \mathop = 1}^\infty a_n$ converges.
$\blacksquare$
Also see
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces