Math 2.0

Real functions Theorem

Triangle inequality for integrals

Builds on Monotonicity of the integralAbsolute convergence

Statement

Let f ⁣:[a,b]→ℝf \colon [a, b] \to \mathbb{R} be Riemann integrable. Then ∣f∣|f| is Riemann integrable on [a,b][a, b], and ∣∫abf(x) dx∣≤∫ab∣f(x)∣ dx.\Big| \int_a^b f(x) \, dx \Big| \le \int_a^b |f(x)| \, dx.

Remarks

The inequality follows from −∣f∣≤f≤∣f∣-|f| \le f \le |f| and the monotonicity of the integral. The converse of the first claim fails. On [0,1][0, 1], let f=1f = 1 on the rationals and f=−1f = -1 elsewhere. Then ∣f∣=1|f| = 1 is integrable, but ff is not. Mathlib states the inequality as intervalIntegral.norm_integral_le_integral_norm.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

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Partition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionRiemann integral, definition, no evidence levelRiemann integraldefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionDefinite integral, definition, no evidence levelDefinite integraldefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionMonotonicity of the integral, statement, no evidence levelMonotonicity of the i…statementAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionTriangle inequality for integrals, statement, no evidence levelTriangle inequality f…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.
  • no evidence level
  1. Partition of an interval definition
  2. Sequence definition
  3. Mesh of a partition definition
  4. Riemann sum definition
  5. Real sequence definition
  6. Riemann integrable function definition
  7. Limit of a sequence definition
  8. Series definition
  9. Riemann integral definition
  10. Convergent sequence definition
  11. Partial sum definition
  12. Definite integral definition
  13. Convergent series definition
  14. Monotonicity of the integral statement
  15. Absolute convergence definition
  16. Triangle inequality for integrals statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Integral, Inequalities
About this page
Tag0NR82KMYKD
URIm2://0NR82KMYKD
Kindstatement, theorem
WritingW0 stub
MSC26A42
StatusThis copy of the site has no build data. Run m2 status 0NR82KMYKD.
UsesMonotonicity of the integral, Absolute convergence

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