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

be a topological vector space be an index set

is called unconditionally convergent if

• is countable, and

• for every permutation (bijection

Alternative definition

converges.

is a Banach space, every absolutely convergent series is unconditionally convergent, but the converse is an infinite-dimensional Banach space, then by Dvoretzky–Rogers theorem by the Riemann series theorem is unconditionally convergent if and only if it is absolutely convergent.

See also

References

• Ch. Heil: A Basis Theory Primer