Unconditional convergence

Unconditional convergence is a topological property (convergence) related to an algebraic object (sum). It is an extension of the notion of convergence for series of countably many elements to series of arbitrarily many. It has been mostly studied in Banach spaces.

A series of numbers is unconditionally convergent if under all reorderings of the numbers, their sum converges to the same value as under the given ordering—their sum is not conditional on the particular arrangement. In contrast, a series is conditionally convergent if it converges but different orderings do not all converge to that same value.

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 $I_{0}:=\{i\in I:x_{i}\neq 0\}$ is countable and
for every permutation of $I_{0}:=\{i\in I:x_{i}\neq 0\}$ the relation holds: $\sum _{{i=1}}^{\infty }x_{i}=x$

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.

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

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Unconditional convergence

Definition

Alternative definition

See also

References