unconditional convergence

A series n=1xn in a Banach space X is unconditionally convergent if for every permutationMathworldPlanetmath σ: the series n=1xσ(n) convergesPlanetmathPlanetmath.

Alternatively, for every chain of finite subsets S1S2 of , the partial sums

kS1xk, kS2xk, ,

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

Yet a third equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Equivalent3) definition is given as follows: A series is unconditionally convergent if for every sequenceMathworldPlanetmath (εn)n=1, with εn{±1}, the series n=1εnxn converges.

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

When X=n then by a famous theorem of Riemann (xn) is unconditionally convergent if and only if it is absolutely convergent.


  • 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