Unconditional convergence

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Unconditional convergence is a topological property (convergence) related to an algebraical 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.


Let X be a topological vector space. Let I be an index set and x_i \in X for all i \in I.

The series \textstyle \sum_{i \in I} x_i is called unconditionally convergent to x \in X, if

Alternative definition[edit]

Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence (\varepsilon_n)_{n=1}^\infty, with \varepsilon_n\in\{-1, +1\}, the series

\sum_{n=1}^\infty \varepsilon_n x_n


Every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general. When X = Rn, then, by the Riemann series theorem, the series \sum x_n is unconditionally convergent if and only if it is absolutely convergent.

See also[edit]


  • Ch. Heil: A Basis Theory Primer
  • K. Knopp: "Theory and application of infinite series"
  • K. Knopp: "Infinite sequences and series"
  • P. Wojtaszczyk: "Banach spaces for analysts"

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