Unconditional convergence

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

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

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

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

Alternative definition

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

\sum _{{n=1}}^{\infty }\varepsilon _{n}x_{n}

converges.

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

References

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.

Unconditional convergence

Definition

Alternative definition

See also

References