Unconditional convergence

Short description: Order-independent convergence of a sequence

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 $\sum _{i\in I}{x}_i$ is called unconditionally convergent to $x\in X,$ if

• the indexing set $I_0:=\{i\in I:x_i\ne 0\}$ is countable, and
• for every permutation (bijection) $\sigma :I_0\to I_0$ of $I_0={\left\{i_k\right\}}_{k=1}^{\mathrm{\infty }}$ the following relation holds: $\sum _{k=1}^{\mathrm{\infty }}{x}_{\sigma \left(i_k\right)}=x.$

Alternative definition

Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence ${\left({\epsilon }_n\right)}_{n=1}^{\mathrm{\infty }},$ with ${\epsilon }_n\in \{-1,+1\},$ the series $$\sum _{n=1}^{\mathrm{\infty }}{\epsilon }_n{x}_n$$ 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 $\sum _n{x}_n$ 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
• Riemann series theorem – Unconditional series converge absolutely

References

• Ch. Heil: A Basis Theory Primer
• Knopp, Konrad (1956). Infinite Sequences and Series. Dover Publications. ISBN 9780486601533. https://archive.org/details/infinitesequence0000knop. 
• 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.