Math 2.0

Sequences, series, summability Theorem

Absolutely convergent series in a Banach space converge

Builds on Banach spaceEpsilon characterisation of Cauchy sequencesComplete metric space

Statement

Let (V,∥⋅∥)(V, \|\cdot\|) be a Banach space and let (xn)n≥1(x_n)_{n \ge 1} be a sequence in VV with ∑n≥1∥xn∥<∞\sum_{n \ge 1} \|x_n\| < \infty. Then the partial sums sN=∑n=1Nxns_N = \sum_{n=1}^{N} x_n converge to a point s∈Vs \in V, and ∥s∥≤∑n≥1∥xn∥\|s\| \le \sum_{n \ge 1} \|x_n\|.

Remarks

Proof: for M>NM > N, the triangle inequality gives ∥sM−sN∥≤∑n=N+1M∥xn∥\|s_M - s_N\| \le \sum_{n=N+1}^{M} \|x_n\|. This tends to 00, so (sN)(s_N) is a Cauchy sequence, and it converges because VV is complete. The converse holds: a normed space in which each absolutely convergent series converges is a Banach space. For V=ℝV = \mathbb{R}, this is the fact that absolutely convergent real series converge.

Reading path

To understand this page from the ground up, read these 37 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 numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSet, 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 spacedefinitionAbsolutely convergent series in a Banach space converge, statement, no evidence levelAbsolutely convergent…statement
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.
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  1. Element definition
  2. Extended real number line object
  3. Proposition concept
  4. Real numbers object
  5. Set bounded above definition
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Predicate definition
  11. Supremum definition
  12. Empty set object
  13. Family of sets definition
  14. Ordered pair definition
  15. Intersection definition
  16. Union definition
  17. Existential quantifier definition
  18. Universal quantifier definition
  19. Cartesian product definition
  20. Subset definition
  21. Union of a family of sets definition
  22. Uniqueness quantifier definition
  23. Binary relation definition
  24. Topological space definition
  25. Function definition
  26. Metric space definition
  27. Convergence of a sequence in a metric space definition
  28. Open ball definition
  29. Diameter of a set definition
  30. Metric topology definition
  31. Epsilon characterisation of Cauchy sequences statement
  32. Complete metric space definition
  33. Open set definition
  34. Equivalent metrics definition
  35. Equivalent norms definition
  36. Banach space definition
  37. Absolutely convergent series in a Banach space converge statement

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Absolute_convergence, Proof that any absolutely convergent series in a Banach space is convergent
  2. Web https://en.wikipedia.org/wiki/Banach_space, Characterization in terms of series
About this page
Tag082MGNWNA8
URIm2://082MGNWNA8
Kindstatement, theorem
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 082MGNWNA8.
UsesBanach space, Epsilon characterisation of Cauchy sequences, Complete metric space

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