Math 2.0

General topology Axiom

Triangle inequality for a metric

Builds on Metric space

Statement

Let (X,d)(X, d) be a metric space. For all points x,y,z∈Xx, y, z \in X, d(x,z)≤d(x,y)+d(y,z).d(x, z) \le d(x, y) + d(y, z).

Remarks

This is one of the four axioms of a metric. For X=ℝX = \mathbb{R} with d(x,y)=∣x−y∣d(x, y) = |x - y|, it is the triangle inequality for real numbers, with a=x−ya = x - y and b=y−zb = y - z. A consequence is the reverse form ∣d(x,y)−d(x,z)∣≤d(y,z)|d(x, y) - d(x, z)| \le d(y, z) for all x,y,z∈Xx, y, z \in X.

Reading path

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

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionTriangle inequality for a metric, 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. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Implication definition
  6. Predicate definition
  7. Ordered pair definition
  8. Existential quantifier definition
  9. Universal quantifier definition
  10. Cartesian product definition
  11. Subset definition
  12. Uniqueness quantifier definition
  13. Binary relation definition
  14. Function definition
  15. Metric space definition
  16. Triangle inequality for a metric statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Triangle_inequality, Metric space
  2. Web https://en.wikipedia.org/wiki/Metric_space, Definition
About this page
Tag01M6CW9VM6
URIm2://01M6CW9VM6
Aliasestriangle-inequality-for-a-metric
Kindstatement, axiom
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 01M6CW9VM6.
UsesMetric space

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