Unconditional convergence
In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge to the same value. In contrast, a series is conditionally convergent if it converges but different orderings do not all converge to that same value. Unconditional convergence is equivalent to absolute convergence in finite-dimensional vector spaces, but is a weaker property in infinite dimensions.
Definition
Let [math]\displaystyle{ X }[/math] be a topological vector space. Let [math]\displaystyle{ I }[/math] be an index set and [math]\displaystyle{ x_i \in X }[/math] for all [math]\displaystyle{ i \in I. }[/math]
The series [math]\displaystyle{ \textstyle \sum_{i \in I} x_i }[/math] is called unconditionally convergent to [math]\displaystyle{ x \in X, }[/math] if
- the indexing set [math]\displaystyle{ I_0 := \left\{i \in I : x_i \neq 0\right\} }[/math] is countable, and
- for every permutation (bijection) [math]\displaystyle{ \sigma : I_0 \to I_0 }[/math] of [math]\displaystyle{ I_0 = \left\{i_k\right\}_{k=1}^\infty }[/math] the following relation holds: [math]\displaystyle{ \sum_{k=1}^\infty x_{\sigma\left(i_k\right)} = x. }[/math]
Alternative definition
Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence [math]\displaystyle{ \left(\varepsilon_n\right)_{n=1}^\infty, }[/math] with [math]\displaystyle{ \varepsilon_n \in \{-1, +1\}, }[/math] the series [math]\displaystyle{ \sum_{n=1}^\infty \varepsilon_n x_n }[/math] converges.
If [math]\displaystyle{ X }[/math] is a Banach space, every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general. Indeed, if [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X = \R^n, }[/math] by the Riemann series theorem, the series [math]\displaystyle{ \sum_n x_n }[/math] is unconditionally convergent if and only if it is absolutely convergent.
See also
- Absolute convergence – Mode of convergence of an infinite series
- Modes of convergence (annotated index) – Annotated index of various modes of convergence
- Riemann series theorem – Unconditional series converge absolutely
References
- Ch. Heil: A Basis Theory Primer
- Knopp, Konrad (1956). Infinite Sequences and Series. Dover Publications. ISBN 9780486601533. https://archive.org/details/infinitesequence0000knop.
- Knopp, Konrad (1990). Theory and Application of Infinite Series. Dover Publications. ISBN 9780486661650.
- Wojtaszczyk, P. (1996). Banach spaces for analysts. Cambridge University Press. ISBN 9780521566759.
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