Unconditional convergence

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

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

Unconditionally 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. 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.

See also

• Modes of convergence (annotated index)

References

• Ch. Heil: A Basis Theory Primer
• K. Knopp: "Theory and application of infinite series"
• K. Knopp: "Infinite sequences and series"
• P. Wojtaszczyk: "Banach spaces for analysts"

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