Series in Banach spaces

Remarks

An absolutely convergent series in a Banach space converges. More generally, if $\norm{x_n} \le b_n$ for all $n$ and $\sum b_n$ converges, then $\sum x_n$ converges; in Mathlib this is Summable.of_norm_bounded. A normed space in which every absolutely convergent series converges is complete, so this property characterizes Banach spaces.

Example: in $\ell^2$, see spaces ℓ^p, let $x_n = e_n / n$ for $n \ge 1$, where $e_n$ is the $n$-th unit vector. Then $\norm{S_N - s}_2 = \left( \sum_{n > N} 1/n^2 \right)^{1/2} \to 0$ for $s = (1, 1/2, 1/3, \dots)$, so the series converges.

Non-example: the same series does not converge absolutely, because $\sum_n \norm{x_n}_2 = \sum_n 1/n$ diverges. This series even converges after every rearrangement. In $\R$ this cannot happen: a real series that converges after every rearrangement converges absolutely.