Math 2.0

Sequences, series, summability Definition

Unconditional convergence

Builds on Normed vector spaceRearrangement of a seriesConvergent series

Definition

Let XX be a normed vector space over ℝ\R or ℂ\C, and let (xn)n≥0(x_n)_{n \ge 0} be a sequence in XX. The series ∑n=0∞xn\sum_{n=0}^{\infty} x_n converges unconditionally if for every bijection σ ⁣:ℕ→ℕ\sigma \colon \N \to \N the rearranged series ∑n=0∞xσ(n)\sum_{n=0}^{\infty} x_{\sigma(n)} converges in XX.

Remarks

When the series converges unconditionally, all rearrangements have the same sum. In a Banach space, unconditional convergence is equivalent to summability of the family (xn)n∈ℕ(x_n)_{n \in \N}. Example: in the sequence space ℓ2\ell^2 with unit vectors ene_n, the series ∑en/n\sum e_n / n converges unconditionally. Indeed, for every finite set FF of indices, ∥∑n∈Fen/n∥2=∑n∈F1/n2\norm{\sum_{n \in F} e_n / n}^2 = \sum_{n \in F} 1/n^2, so the partial sums of every rearrangement are Cauchy. This series does not converge absolutely, since ∑∥en/n∥=∑1/n\sum \norm{e_n / n} = \sum 1/n diverges. Non-example: the alternating harmonic series converges, but some rearrangements diverge, by the Riemann rearrangement theorem. In ℝd\R^d, a series converges unconditionally if and only if it converges absolutely.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Sequence, definition, no evidence levelSequencedefinitionVector space, definition, no evidence levelVector spacedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionNorm, definition, no evidence levelNormdefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionNormed vector space, definition, no evidence levelNormed vector spacedefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionRearrangement of a series, definition, no evidence levelRearrangement of a se…definitionConvergent series, definition, no evidence levelConvergent seriesdefinitionUnconditional convergence, definition, no evidence levelUnconditional converg…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Sequence definition
  2. Vector space definition
  3. Real sequence definition
  4. Norm definition
  5. Limit of a sequence definition
  6. Series definition
  7. Normed vector space definition
  8. Convergent sequence definition
  9. Partial sum definition
  10. Rearrangement of a series definition
  11. Convergent series definition
  12. Unconditional convergence definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Unconditional_convergence
About this page
TagGA7Q813F50
URIm2://GA7Q813F50
Kinddefinition
WritingW0 stub
MSC40A05, 46B15
DefinesUnconditional convergence
UsesNormed vector space, Rearrangement of a series, Convergent series
Changed2 November 2026

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