Unconditional convergence

From Wikipedia, the free encyclopedia

In mathematical analysis, a series $\sum_{n=1}^\infty x_n$ in a Banach space X is unconditionally convergent if for every permutation $\sigma: \mathbb N \to \mathbb N$ the series $\sum_{n=1}^\infty x_{\sigma(n)}$ 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 $\varepsilon_n\in\{\pm 1\}$ , 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.