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

Alternative definition

Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence $\left(\varepsilon _{n}\right)_{n=1}^{\infty },$ 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 $X=\mathbb {R} ^{n},$ by the Riemann series theorem, the series ${\textstyle \sum _{n}x_{n}}$ is unconditionally convergent if and only if it is absolutely convergent.

See also

• Absolute convergence – Mode of convergence of an infinite series
• Modes of convergence (annotated index) – Annotated index of various modes of convergence
• Rearrangements and unconditional convergence/Dvoretzky–Rogers theorem – Mode of convergence of an infinite series
• Riemann series theorem – Unconditionally convergent series converge absolutely

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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