Unconditional convergence

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In mathematical analysis, a series \sum_{n=1}^\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}^\infty x_{\sigma(n)} converges.

This notion 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\{\pm 1\}, the series

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

converges.

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

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

[edit] References

  • 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 Uncoditional convergence on PlanetMath, which is licensed under the GFDL.

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