Let $V$ be a normed space with norm $\|\|$. $\sum_{n=1}^\infty a_n$ is an absolutely convergent series if $\sum_{n=1}^\infty \|a_n\|$ converges.
Could you please explain why in complete normed spaces the absolute convergence implies convergence, but it doesn't hold for incomplete normed spaces?
Best Answer
This is the proof of the fact that, if $V$ is a complete normed space, then every absolutely convergent series is convergent. You will see that we need through the proof the fact that $V$ is complete.
Let $\sum_{n=1}^\infty \lVert a_n\rVert$ be a convergent series, which means that $\sum_{n=1}^\infty a_n$ is absolutely convergent. We want to prove that this implies that $\sum_{n=1}^\infty a_n$ is a convergent series. But by definition of a convergent series, it will be convergent if and only if the sequence $\{S_k\}_{k=1}^\infty=\{\sum_{n=1}^k a_n\}_{k=1}^\infty$ is convergent. Since $V$ is complete, this sequence will be convergent if and only if it is a Cauchy sequence. (What we can prove is that this will always be a Cauchy sequence, but if the space $V$ was not complete, we couldn't conclude from here that it would be convergent).
And indeed, it is a Cauchy sequence because:
$$\lVert S_k-S_p\rVert=\biggl\lVert\sum_{n=p+1}^k a_n\biggr\rVert\leq\sum_{n=p+1}^k \lVert a_n\rVert$$
and this expression tends to zero when $k$ and $p$ are taken as close as necessary. This way we have proven that the series $\sum_{n=1}^\infty a_n$ is convergent, as we needed to show.