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

The series $\textstyle \sum_ x_i$ is called unconditionally convergent to $x \in X,$ if
* the indexing set $I_0 := \left\$ is countable, and
* for every permutation ( bijection) $\sigma : I_0 \to I_0$ of $I_0 = \left\_^\infty$ the following relation holds: $\sum_^\infty x_ = x.$

Alternative definition

Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence $\left(\varepsilon_n\right)_^\infty,$ with $\varepsilon_n \in \,$ the series
$\sum_^\infty \varepsilon_n x_n$
converges.

If $X$ is a Banach space, every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general. Indeed, 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 $X = \R^n,$ by the Riemann series theorem, the series $\sum_n x_n$ is unconditionally convergent if and only if it is absolutely convergent.

See also

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References

* Ch. Heil
A Basis Theory Primer
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{{PlanetMath attribution, urlname=unconditionalconvergence, title=Unconditional convergence

Convergence (mathematics)
Mathematical analysis
Mathematical series
Summability theory