A series $\displaystyle{\sum_{n=1}^\infty x_n}$ in a Banach space $X$ is unconditionally convergent if for every permutation $\sigma: \sN \to \sN$ the series $\displaystyle{\sum_{n=1}^\infty x_{\sigma(n)}}$ converges.

Alternatively, for every chain of finite subsets $S_1\subseteq S_2\subseteq\cdots$ of $\mathbb{N}$ , the partial sums $$\sum_{k\in S_1} x_k,\mbox{ }\sum_{k\in S_2} x_k,\mbox{ },\ldots$$ converges. The trick to see this equivalence is to realize two facts: 1. every subsequence of a convergent sequence is convergent, and 2. every chain $\lbrace S_i\rbrace$ can be enlarged to a maximal chain $\lbrace T_i\rbrace$ , such that $|T_i|=i$ . Then the series indexed by $\{S_i\}$ is a subseries indexed by $\{T_i\}$ , which is a subseries of a permutation of the original convergent series.

Yet a third equivalent definition is given as follows: A series is unconditionally convergent if for every sequence $(\varepsilon_n)_{n=1}^\infty$ , with $\varepsilon_n\in\{\pm 1\}$ , the series $\displaystyle{\sum_{n=1}^\infty \varepsilon_n x_n}$ converges.

Every absolutely convergent series is unconditionally convergent, the converse implication does not hold in general.

When $X=\sR^n$ then by a famous theorem of Riemann $(\sum x_n)$ is unconditionally convergent if and only if it is absolutely convergent.

Bibliography

1

K. Knopp: Theory and application of infinite series.

2

K. Knopp: Infinite sequences and series.

3

P. Wojtaszczyk: Banach spaces for analysts.

4

Ch. Heil: A basis theory primer.