Unconditional convergence explained

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

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

• the indexing set

I _{0} : = \left\{i \in I : x _{i} ≠ 0 \right\} is countable, and

• for every permutation (bijection)

\sigma : I _{0} \to I _{0} of I _{0} = \left\{i _{k\right\}} ^{inf ty} ^{k =1} the following relation holds: \sum ^{inf ty} _{k =1} 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)} ^{inf ty,} ^{n =1} with \varepsilon _{n} \in \{ -1 , +1 \}, the series$$\backslash sum\_^\backslash infty\; \backslash varepsilon\_n\; x\_n$$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 = \R ^{n,} by the Riemann series theorem, the series $\backslash sum\_n\; x\_n$ is unconditionally convergent if and only if it is absolutely convergent.

References

• Ch. Heil: A Basis Theory Primer
• Book: Knopp , Konrad . Infinite Sequences and Series. registration. 9780486601533. Dover Publications. 1956.
• Book: Knopp , Konrad . Theory and Application of Infinite Series. Dover Publications. 1990. 9780486661650.
• Book: Wojtaszczyk , P. . Banach spaces for analysts. 1996. Cambridge University Press. 9780521566759.