Unconditional convergence
In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge. 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 be a topological vector space. Let be an index set and for all.
The series is called unconditionally convergent to , if
Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence, with, the series
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 = Rn, by the Riemann series theorem, the series is unconditionally convergent if and only if it is absolutely convergent.
