Unconditional convergence

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Unconditional convergence is a topological property (convergence) related to an algebraic object (sum). It is an extension of the notion of convergence for series of countably many elements to series of arbitrarily many. It has been mostly studied in Banach spaces.

A series of numbers is unconditionally convergent if under all reorderings of the numbers, their sum converges to the same value as under the given ordering—their sum is not conditional on the particular arrangement. In contrast, a series is conditionally convergent if it converges but different orderings do not all converge to that same value.


Let be a topological vector space. Let be an index set and for all .

The series is called unconditionally convergent to , if

  • the indexing set is countable and
  • for every permutation of the relation holds:

Alternative definition[edit]

Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence , with , the series


Every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general: if X is an infinite dimensional Banach space, then by Dvoretzky–Rogers theorem there always exists an unconditionally convergent series in this space that is not absolutely convergent. However, when X = Rn, then, by the Riemann series theorem, the series is unconditionally convergent if and only if it is absolutely convergent.

See also[edit]


This article incorporates material from Unconditional convergence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.