---
tag: 4670W2WNFX
kind: definition
name: Series in Banach spaces
msc: [40A05, 46B15]
defines: WN6M8B3ZKB
uses: [0BDTTM24DP, 0NB643K60Q, 0CVVKY7GY2, 082MGNWNA8, 08P6T2WH6R]
sources:
  - url: https://en.wikipedia.org/wiki/Absolute_convergence
---

## Definition

Let $X$ be a [[Banach space|0BDTTM24DP]] over $\R$ or $\C$ with norm $\norm{\cdot}$, and let $(x_n)_{n \ge 0}$ be a sequence in $X$.
The [[partial sums|0NB643K60Q]] of the series $\sum_n x_n$ are $S_N = \sum_{n=0}^{N} x_n$.
The series *converges* to $s \in X$, written $\sum_{n \ge 0} x_n = s$, if $\norm{S_N - s} \to 0$ as $N \to \infty$.
The series *converges absolutely* if the real series $\sum_{n \ge 0} \norm{x_n}$ [[converges|0CVVKY7GY2]].

## Remarks

[[An absolutely convergent series in a Banach space converges|082MGNWNA8]].
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|08P6T2WH6R]], 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.