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# unconditional convergence

A series $\displaystyle{\sum_{{n=1}}^{\infty}x_{n}}$ in a Banach space $X$ is *unconditionally convergent*
if for every permutation $\sigma:\mathbb{N}\to\mathbb{N}$ the series $\displaystyle{\sum_{{n=1}}^{\infty}x_{{\sigma(n)}}}$
converges.

Alternatively, for every chain of finite subsets $S_{1}\subseteq S_{2}\subseteq\cdots$ of $\mathbb{N}$, the partial sums

$\sum_{{k\in S_{1}}}x_{k},\mbox{ }\sum_{{k\in S_{2}}}x_{k},\mbox{ },\ldots$ |

converges. The trick to see this equivalence is to realize two facts: 1. every subsequence of a convergent sequence is convergent, and 2. every chain $\{S_{i}\}$ can be enlarged to a maximal chain $\{T_{i}\}$, such that $|T_{i}|=i$. Then the series indexed by $\{S_{i}\}$ is a subseries indexed by $\{T_{i}\}$, which is a subseries of a permutation of the original convergent series.

Yet a third equivalent definition is given as follows: A series is unconditionally convergent if for every sequence $(\varepsilon_{n})_{{n=1}}^{\infty}$, with $\varepsilon_{n}\in\{\pm 1\}$, the series $\displaystyle{\sum_{{n=1}}^{\infty}\varepsilon_{n}x_{n}}$ converges.

Every absolutely convergent series is unconditionally convergent, the converse implication does not hold in general.

When $X=\mathbb{R}^{n}$ then by a famous theorem of Riemann $(\sum x_{n})$ is unconditionally convergent if and only if it is absolutely convergent.

# References

- 1 K. Knopp: Theory and application of infinite series.
- 2 K. Knopp: Infinite sequences and series.
- 3 P. Wojtaszczyk: Banach spaces for analysts.
- 4 Ch. Heil: A basis theory primer.

## Mathematics Subject Classification

40A05*no label found*

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