Unconditional convergence

In mathematical analysis, a series $sum\_\{n=1\}^infty\; x\_n$ in a Banach space "X" is unconditionally convergent if for every permutation $sigma:\; mathbb\; N\; o\; mathbb\; N$ the series $sum\_\{n=1\}^inftyx\_\{sigma(n)\}$ converges.

This notion is often defined in an equivalent way: A series is unconditionally convergent if for every sequence $(varepsilon\_n)\_\{n=1\}^infty$, with $varepsilon\_nin\{pm\; 1\}$, the series

:$sum\_\{n=1\}^infty\; varepsilon\_n\; x\_n$

converges.

Every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general. When $X=mathbb\; R^n$ then by the Riemann series theorem, $(x\_n)$ is unconditionally convergent if and only if it is absolutely convergent.

ee also

*Modes of convergence (annotated index)

References

* Ch. Heil: [http://www.math.gatech.edu/~heil/papers/bases.pdf A Basis Theory Primer]
* K. Knopp: "Theory and application of infinite series"
* K. Knopp: "Infinite sequences and series"
* P. Wojtaszczyk: "Banach spaces for analysts"