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

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

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

Alternative definition

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

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

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 ${\displaystyle \sum x_{n}}$ is unconditionally convergent if and only if it is absolutely convergent.

See also

• Modes of convergence (annotated index)
• Riemann series theorem
• Dvoretzky–Rogers theorem
• Rearrangements and unconditional convergence

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.

This article incorporates material from Unconditional convergence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.