Math 2.0

definition

Normal convergence

Definition

Let SS be a set and let fn ⁣:S→Rf_n \colon S \to \mathbb{R} be bounded functions for n≥0n \ge 0. The series of functions ∑n=0∞fn\sum_{n=0}^{\infty} f_n converges normally on SS if ∑n=0∞sup⁡x∈S∣fn(x)∣<∞.\sum_{n=0}^{\infty} \sup_{x \in S} |f_n(x)| < \infty.

Remarks

By the Weierstrass M-test, a normally convergent series converges absolutely and uniformly on SS. The converse fails. For constant functions fn=(−1)n/(n+1)f_n = (-1)^n / (n + 1), the series converges uniformly but not normally. René Baire introduced the concept in 1908.

Explanations

L1 human

Normal convergence means that the sizes of the functions, measured by their suprema, form a convergent series. It is a simple condition that is stronger than uniform convergence.

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To read this node, read these 25 nodes in this order. The foundations come first, and the list ends with this node.

Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionCauchy sequence, definition, no evidence levelCauchy sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionCauchy sequences are bounded, statement, no evidence levelCauchy sequences are …statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementPointwise convergence, definition, no evidence levelPointwise convergencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementConvergent series, definition, no evidence levelConvergent seriesdefinitionUniform convergence, definition, no evidence levelUniform convergencedefinitionCauchy criterion for sequences, statement, no evidence levelCauchy criterion for …statementUniform convergence on a set, definition, no evidence levelUniform convergence o…definitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementSeries of functions, definition, no evidence levelSeries of functionsdefinitionUniformly Cauchy sequence of functions, definition, no evidence levelUniformly Cauchy sequ…definitionComparison test, statement, no evidence levelComparison teststatementWeierstrass M-test, statement, no evidence levelWeierstrass M-teststatementNormal convergence, definition, no evidence levelNormal convergencedefinition
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  1. Sequence definition
  2. Real sequence definition
  3. Subsequence definition
  4. Bounded sequence definition
  5. Cauchy sequence definition
  6. Limit of a sequence definition
  7. Monotone sequence definition
  8. Series definition
  9. Cauchy sequences are bounded statement
  10. Monotone convergence theorem for sequences statement
  11. Convergent sequence definition
  12. Every sequence has a monotone subsequence statement
  13. Pointwise convergence definition
  14. Partial sum definition
  15. Bolzano–Weierstrass theorem statement
  16. Convergent series definition
  17. Uniform convergence definition
  18. Cauchy criterion for sequences statement
  19. Uniform convergence on a set definition
  20. A series of nonnegative terms converges iff its partial sums are bounded statement
  21. Series of functions definition
  22. Uniformly Cauchy sequence of functions definition
  23. Comparison test statement
  24. Weierstrass M-test statement
  25. Normal convergence definition

Sources

  1. Web https://en.wikipedia.org/wiki/Normal_convergence, Definition

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