---
tag: GA7Q813F50
kind: definition
name: Unconditional convergence
msc: [40A05, 46B15]
defines: 2ME9M61TJK
uses: [00A0RPAQ8A, 0HRJ2Y74T3, 0CVVKY7GY2]
sources:
  - url: https://en.wikipedia.org/wiki/Unconditional_convergence
---

## Definition

Let $X$ be a [[normed vector space|00A0RPAQ8A]] over $\R$ or $\C$, and let $(x_n)_{n \ge 0}$ be a sequence in $X$.
The series $\sum_{n=0}^{\infty} x_n$ converges unconditionally if for every bijection $\sigma \colon \N \to \N$ the [[rearranged series|0HRJ2Y74T3]] $\sum_{n=0}^{\infty} x_{\sigma(n)}$ [[converges|0CVVKY7GY2]] in $X$.

## Remarks

When the series converges unconditionally, all rearrangements have the same sum.
In a Banach space, unconditional convergence is equivalent to [[summability of the family|70HJTK5VZH]] $(x_n)_{n \in \N}$.
Example: in the [[sequence space|08P6T2WH6R]] $\ell^2$ with unit vectors $e_n$, the series $\sum e_n / n$ converges unconditionally.
Indeed, for every finite set $F$ of indices, $\norm{\sum_{n \in F} e_n / n}^2 = \sum_{n \in F} 1/n^2$, so the partial sums of every rearrangement are Cauchy.
This series does not converge absolutely, since $\sum \norm{e_n / n} = \sum 1/n$ diverges.
Non-example: the [[alternating harmonic series|0JQ5Y3WWVE]] converges, but some rearrangements diverge, by the [[Riemann rearrangement theorem|0E7VBYGFF7]].
In $\R^d$, a series converges unconditionally if and only if it converges [[absolutely|0W3K3QPEWT]].

## Explanations

### L1 human

Unconditional convergence asks that the sum not depend on the order of the terms.
In finite dimensions this is the same as absolute convergence, but in spaces of infinite dimension it is weaker.