# unconditional convergence

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

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

$$\sum _{k\in {S}_{1}}{x}_{k},\text{}\sum _{k\in {S}_{2}}{x}_{k},\text{},\mathrm{\dots}$$ |

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^{} ${({\epsilon}_{n})}_{n=1}^{\mathrm{\infty}}$, with ${\epsilon}_{n}\in \{\pm 1\}$, the
series $\sum _{n=1}^{\mathrm{\infty}}}{\epsilon}_{n}{x}_{n$ 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 |
---|---|

Canonical name | UnconditionalConvergence |

Date of creation | 2013-03-22 15:29:57 |

Last modified on | 2013-03-22 15:29:57 |

Owner | kompik (10588) |

Last modified by | kompik (10588) |

Numerical id | 11 |

Author | kompik (10588) |

Entry type | Definition |

Classification | msc 40A05 |

Synonym | unconditionally convergent |

Related topic | AbsoluteConvergence |

Related topic | ConditionallyConvergentSeriesOfRealNumbersCanBeRearrangedToConvergeToAnyNumber |