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

Alternative definition

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

If ${\displaystyle X}$ is a Banach space, every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general. Indeed, if ${\displaystyle X}$ 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 ${\displaystyle X=\mathbb {R} ^{n},}$ 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