Let ${\displaystyle X}$  be a topological vector space. Let ${\displaystyle I}$  be an index set and ${\displaystyle x_{i}\in X}$  for all ${\displaystyle i\in I.}$ 

The series ${\displaystyle \textstyle \sum _{i\in I}x_{i}}$  is called unconditionally convergent to ${\displaystyle x\in X,}$  if

• the indexing set ${\displaystyle I_{0}:=\left\{i\in I:x_{i}\neq 0\right\}}$  is countable, and
• for every permutation (bijection) ${\displaystyle \sigma :I_{0}\to I_{0}}$  of ${\displaystyle I_{0}=\left\{i_{k}\right\}_{k=1}^{\infty }}$  the following relation holds: ${\displaystyle \sum _{k=1}^{\infty }x_{\sigma \left(i_{k}\right)}=x.}$ 

Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence ${\displaystyle \left(\varepsilon _{n}\right)_{n=1}^{\infty },}$  with ${\displaystyle \varepsilon _{n}\in \{-1,+1\},}$  the series

If ${\displaystyle X}$  is a Banach space, every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general. Indeed, if ${\displaystyle X}$  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 ${\displaystyle X=\mathbb {R} ^{n},}$  by the Riemann series theorem, the series ${\textstyle \sum _{n}x_{n}}$  is unconditionally convergent if and only if it is absolutely convergent.

• Absolute convergence – Mode of convergence of an infinite series
• Modes of convergence (annotated index) – Annotated index of various modes of convergence
• Rearrangements and unconditional convergence/Dvoretzky–Rogers theorem – Mode of convergence of an infinite series
• Riemann series theorem – Unconditional series converge absolutely

• Ch. Heil: A Basis Theory Primer
• Knopp, Konrad (1956). Infinite Sequences and Series. Dover Publications. ISBN 9780486601533.
• 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.

This article incorporates material from Unconditional convergence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.