DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-14

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Unconditional convergence of a Banach-space series

Definition

Let $X$ be a normed space and let $x : N_{\geq 1} \to X$ be a positively indexed family. The notation $\sum_{n = 1}^{\infty} x_{n}$ denotes the zero-indexed series whose term at $m \in N$ is $x_{m + 1}$ . This series is unconditionally convergent to $s \in X$ if for every permutation $π : N_{\geq 1} \to N_{\geq 1}$ , the rearranged series $\sum_{n = 1}^{\infty} x_{π (n)}$ , interpreted by the same shift, converges in norm to $s$ .

This is different from absolute convergence, which means $\sum_{n} ∥ x_{n} ∥ < \infty$ . The fixed-order series, all rearrangements, and the scalar series of norms are therefore kept distinct.

Depends on

Series and absolute convergence in a normed space

Used by

Unconditional and conditional Schauder bases Definition
Equivalent forms of unconditional convergence Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Michael Müger, Introduction to Functional Analysis (standard reference, not scraped)