Math 2.0

Sequences, series, summability Absolute and conditional convergence, rearrangements, products of series Theorem

Lévy–Steinitz theorem

Builds on Rearrangement of a seriesAffine subspaceEuclidean spaceDual spaceAbsolute convergence

Statement

Let d≥1d \ge 1 and let (xn)n≥0(x_n)_{n \ge 0} be a sequence in the Euclidean space ℝd\R^d. Let SS be the set of all s∈ℝds \in \R^d such that some rearrangement ∑nxσ(n)\sum_n x_{\sigma(n)} converges to ss. Let Γ\Gamma be the set of linear functionals f ⁣:ℝd→ℝf \colon \R^d \to \R such that ∑n∣f(xn)∣<∞\sum_n \abs{f(x_n)} < \infty. If SS is not empty, then S=s0+{v∈ℝd:f(v)=0 for all f∈Γ}S = s_0 + \set{v \in \R^d : f(v) = 0 \text{ for all } f \in \Gamma} for any s0∈Ss_0 \in S. In particular, SS is either empty or an affine subspace of ℝd\R^d.

Remarks

For d=1d = 1 this is the Riemann rearrangement theorem. If the series converges absolutely, then Γ\Gamma contains all functionals and SS is a single point. If it converges conditionally, then Γ={0}\Gamma = \set{0} and S=ℝS = \R. Example in ℝ2\R^2: take xn=((−1)n/(n+1),1/2n)x_n = ((-1)^n/(n+1), 1/2^n). The second coordinate converges absolutely, the first only conditionally, so SS is a horizontal line {(t,2):t∈ℝ}\set{(t, 2) : t \in \R}. The proof is harder than in dimension one. Its core is a lemma of Steinitz on finitely many vectors in ℝd\R^d with sum zero. They can be ordered so that every partial sum has norm at most CdC_d times the largest norm of the vectors, where CdC_d depends only on dd. In infinite-dimensional Banach spaces the conclusion can fail: the set of sums need not be an affine subspace.

Going deeper

For a student

The theorem extends Riemann's rearrangement theorem from ℝ\R to ℝd\R^d. The subspace of possible shifts is cut out by the functionals that make the series absolutely convergent. It explains why unconditional convergence and absolute convergence coincide in ℝd\R^d: the set SS is a single point exactly when every functional is in Γ\Gamma.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

Show the path as a graph
Affine space, concept, no evidence levelAffine spaceconceptEuclidean space, object, no evidence levelEuclidean spaceobjectSequence, definition, no evidence levelSequencedefinitionVector space, definition, no evidence levelVector spacedefinitionAffine subspace, definition, no evidence levelAffine subspacedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionLinear map, definition, no evidence levelLinear mapdefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionLinear form, definition, no evidence levelLinear formdefinitionSpace of linear maps, definition, no evidence levelSpace of linear mapsdefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionRearrangement of a series, definition, no evidence levelRearrangement of a se…definitionDual space, definition, no evidence levelDual spacedefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionLévy–Steinitz theorem, statement, no evidence levelLévy–Steinitz theoremstatement
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. Affine space concept
  2. Euclidean space object
  3. Sequence definition
  4. Vector space definition
  5. Affine subspace definition
  6. Real sequence definition
  7. Linear map definition
  8. Limit of a sequence definition
  9. Series definition
  10. Linear form definition
  11. Space of linear maps definition
  12. Convergent sequence definition
  13. Partial sum definition
  14. Rearrangement of a series definition
  15. Dual space definition
  16. Convergent series definition
  17. Absolute convergence definition
  18. Lévy–Steinitz theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/L%C3%A9vy%E2%80%93Steinitz_theorem
About this page
Tag6EXHAQF8SP
URIm2://6EXHAQF8SP
Kindstatement, theorem
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 6EXHAQF8SP.
AboutRearrangement of a series
UsesRearrangement of a series, Affine subspace, Euclidean space, Dual space, Absolute convergence
Same as
Changed2 November 2026

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