Math 2.0

Functional analysis Theorem

Orlicz–Pettis theorem

Builds on Banach spaceUnconditional convergenceSeries in Banach spaces

Statement

Let XX be a Banach space over ℝ\R or ℂ\C, and let X′X' be the space of continuous linear functionals on XX. Let (xn)n≥1(x_n)_{n \ge 1} be a sequence in XX. Assume that for every strictly increasing sequence n1<n2<⋯n_1 < n_2 < \cdots of indices there is y∈Xy \in X with φ(y)=∑k=1∞φ(xnk)for all φ∈X′.\varphi(y) = \sum_{k=1}^{\infty} \varphi(x_{n_k}) \quad \text{for all } \varphi \in X'. Then for every such sequence of indices the series ∑kxnk\sum_{k} x_{n_k} converges in norm. In particular, the series ∑nxn\sum_n x_n is unconditionally convergent in norm.

Remarks

The hypothesis says that every subseries converges in the weak topology; the conclusion says that every subseries converges in norm. The theorem needs all subseries: in ℓ2\ell^2, the series e1+(e2−e1)+(e3−e2)+⋯e_1 + (e_2 - e_1) + (e_3 - e_2) + \cdots has partial sums ene_n, which tend to 00 weakly but not in norm. The theorem also holds in Hausdorff locally convex spaces, with the weak topology given by the continuous dual.

Going deeper

For a student

The theorem is a basic tool for series in Banach spaces and for vector measures. A countably additive vector measure is defined by convergence of ∑μ(An)\sum \mu(A_n) for disjoint sets AnA_n, and every subfamily of disjoint sets gives a subseries. So the theorem shows that a weakly countably additive vector measure is countably additive in norm. It is also used in the Bochner and Pettis theories of integration of vector-valued functions.

Reading path

To understand this page from the ground up, read these 39 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionExtended real number line, object, no evidence levelExtended real number …objectProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSeries in Banach spaces, concept, no evidence levelSeries in Banach spac…conceptSet bounded above, definition, no evidence levelSet bounded abovedefinitionUnconditional convergence, concept, no evidence levelUnconditional converg…conceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionSupremum, definition, no evidence levelSupremumdefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionConvergence of a sequence in a metric space, definition, no evidence levelConvergence of a sequ…definitionOpen ball, definition, no evidence levelOpen balldefinitionDiameter of a set, definition, no evidence levelDiameter of a setdefinitionMetric topology, definition, no evidence levelMetric topologydefinitionEpsilon characterisation of Cauchy sequences, statement, no evidence levelEpsilon characterisat…statementComplete metric space, definition, no evidence levelComplete metric spacedefinitionOpen set, definition, no evidence levelOpen setdefinitionEquivalent metrics, definition, no evidence levelEquivalent metricsdefinitionEquivalent norms, definition, no evidence levelEquivalent normsdefinitionBanach space, definition, no evidence levelBanach spacedefinitionOrlicz–Pettis theorem, statement, no evidence levelOrlicz–Pettis 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. Element definition
  2. Extended real number line object
  3. Proposition concept
  4. Real numbers object
  5. Series in Banach spaces concept
  6. Set bounded above definition
  7. Unconditional convergence concept
  8. Set definition
  9. Conjunction definition
  10. Disjunction definition
  11. Implication definition
  12. Predicate definition
  13. Supremum definition
  14. Empty set object
  15. Family of sets definition
  16. Ordered pair definition
  17. Intersection definition
  18. Union definition
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. Cartesian product definition
  22. Subset definition
  23. Union of a family of sets definition
  24. Uniqueness quantifier definition
  25. Binary relation definition
  26. Topological space definition
  27. Function definition
  28. Metric space definition
  29. Convergence of a sequence in a metric space definition
  30. Open ball definition
  31. Diameter of a set definition
  32. Metric topology definition
  33. Epsilon characterisation of Cauchy sequences statement
  34. Complete metric space definition
  35. Open set definition
  36. Equivalent metrics definition
  37. Equivalent norms definition
  38. Banach space definition
  39. Orlicz–Pettis theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Orlicz%E2%80%93Pettis_theorem
About this page
TagNFCQYZ3GRM
URIm2://NFCQYZ3GRM
Kindstatement, theorem
WritingW0 stub
MSC46B15, 40A05
StatusThis copy of the site has no build data. Run m2 status NFCQYZ3GRM.
AboutSeries in Banach spaces, Unconditional convergence
UsesBanach space, Unconditional convergence, Series in Banach spaces
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Changed2 November 2026

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