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 (http://planetmath.org/Equivalent3) 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: http://www.math.gatech.edu/ heil/papers/bases.pdfA basis theory primer.
Title unconditional convergence UnconditionalConvergence 2013-03-22 15:29:57 2013-03-22 15:29:57 kompik (10588) kompik (10588) 11 kompik (10588) Definition msc 40A05 unconditionally convergent AbsoluteConvergence ConditionallyConvergentSeriesOfRealNumbersCanBeRearrangedToConvergeToAnyNumber