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 $\sum_{i \in I} x_{i}$ 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 (bijection) $σ : I_{0} \to I_{0}$ of $I_{0} = {i_{k}}_{k = 1}^{\infty}$ the following relation holds: $\sum_{k = 1}^{\infty} x_{σ (i_{k})} = x .$

Alternative definition

Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence ${(ε_{n})}_{n = 1}^{\infty},$ with $ε_{n} \in {- 1, + 1},$ the series $\sum_{n = 1}^{\infty} ε_{n} x_{n}$ 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 $\sum_{n} x_{n}$ is unconditionally convergent if and only if it is absolutely convergent.

References

Ch. Heil: A Basis Theory Primer
Knopp, Konrad (1956). Infinite Sequences and Series. Dover Publications. ISBN 9780486601533. {{cite book}}: ISBN / Date incompatibility (help)
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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v t e Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector-valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function Measure theory in topological vector spaces
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold

Unconditional convergence

Definition

Alternative definition

See also

References