Absolutely Convergent Series is Convergent iff Normed Vector Space is Banach/Necessary Condition

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Theorem

Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space.

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $X$.

Suppose $X$ is a Banach space.


Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is convergent.

Proof

That $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is absolutely convergent means that $\displaystyle \sum_{n \mathop = 1}^\infty \norm {a_n}$ converges in $\R$.

Hence the sequence of partial sums is a Cauchy sequence by Convergent Sequence is 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:

$\displaystyle \sum_{k \mathop = n + 1}^m \norm {a_k} = \size {\sum_{k \mathop = 1}^m \norm {a_k} - \sum_{k \mathop = 1}^n \norm {a_k} } < \epsilon$

This $N$ exists because the sequence is Cauchy.


Now observe that, for $m \ge n \ge N$, one also has:

\(\displaystyle \norm {\sum_{k \mathop = 1}^m a_k - \sum_{k \mathop = 1}^n a_k}\) \(=\) \(\displaystyle \norm {\sum_{k \mathop = n + 1}^m a_k}\)
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{k \mathop = n + 1}^m \norm {a_k}\) Triangle inequality for $\norm {\, \cdot \,}$
\(\displaystyle \) \(<\) \(\displaystyle \epsilon\)

It follows that the sequence of partial sums of $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is Cauchy.

As $X$ is a Banach space, this implies that $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges.

$\blacksquare$

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