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Concept
Unconditional convergence
Summary
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.
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 (bijection) of the following relation holds:
Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence with the series
converges.
If is a Banach space, every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general. Indeed, if 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 by the Riemann series theorem, the series is unconditionally convergent if and only if it is absolutely convergent.
Official source
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