PlanetMath: unconditional convergence

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unconditional convergence

(Definition)

A series $\displaystyle{\sum_{n=1}^\infty x_n}$ in a Banach space 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

$\displaystyle \sum_{k\in S_1} x_k,$ $\displaystyle \sum_{k\in S_2} x_k,$ $\displaystyle ,\ldots$

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 $\lbrace S_i\rbrace$ can be enlarged to a maximal chain $\lbrace T_i\rbrace$ , such that $\vert T_i\vert=i$ . 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.

Bibliography

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.

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"unconditional convergence" is owned by kompik. [ full author list (3) ]

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Other names:

unconditionally convergent

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Cross-references: implication, converse, absolutely convergent, sequence, convergent series, indexed by, convergent, convergent sequence, subsequence, equivalence, partial sums, subsets, finite, chain, converges, permutation, Banach space, series
There is 1 reference to this entry.

This is version 8 of unconditional convergence, born on 2005-09-05, modified 2007-01-13.
Object id is 7358, canonical name is UncoditionalConvergence.
Accessed 2375 times total.

Classification:

AMS MSC:

40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)

Pending Errata and Addenda

None.

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