Math 2.0

General topology Theorem

Completeness of the space of bounded continuous functions

Builds on Complete metric spaceContinuous map of topological spacesUniform metric on bounded functions

Statement

Let XX be a topological space and let (Y,dY)(Y, d_Y) be a complete metric space. Let Cb(X,Y)C_b(X, Y) be the set of continuous maps f ⁣:X→Yf \colon X \to Y whose image f(X)f(X) is bounded in YY. For f,g∈Cb(X,Y)f, g \in C_b(X, Y), let d∞(f,g)=sup⁡x∈XdY(f(x),g(x))d_\infty(f, g) = \sup_{x \in X} d_Y(f(x), g(x)), with d∞(f,g)=0d_\infty(f, g) = 0 when XX is empty. Then (Cb(X,Y),d∞)(C_b(X, Y), d_\infty) is a complete metric space.

Remarks

For Y=ℝY = \mathbb{R}, the metric d∞d_\infty is the uniform metric, and it comes from the sup norm. Proof sketch: a Cauchy sequence (fn)(f_n) in d∞d_\infty is Cauchy at each point xx, so it has a pointwise limit f(x)f(x) in YY. The convergence is uniform, and a uniform limit of continuous maps is continuous. The limit is also bounded. For a compact space XX, each continuous map to ℝ\mathbb{R} is bounded, so C(X,ℝ)C(X, \mathbb{R}) is complete with the sup metric.

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 abovedefinitionUniform metric on bounded functions, example, no evidence levelUniform metric on bou…exampleSet, 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 levelFunctiondefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionMetric 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…statementOpen set, definition, no evidence levelOpen setdefinitionComplete metric space, definition, no evidence levelComplete metric spacedefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionCompleteness of the space of bounded continuous functions, statement, no evidence levelCompleteness of the s…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. Uniform metric on bounded functions example
  7. Set definition
  8. Conjunction definition
  9. Disjunction definition
  10. Implication definition
  11. Predicate definition
  12. Supremum definition
  13. Empty set object
  14. Family of sets definition
  15. Ordered pair definition
  16. Intersection definition
  17. Union definition
  18. Existential quantifier definition
  19. Universal quantifier definition
  20. Cartesian product definition
  21. Subset definition
  22. Union of a family of sets definition
  23. Uniqueness quantifier definition
  24. Binary relation definition
  25. Topological space definition
  26. Function definition
  27. Preimage of a set 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. Open set definition
  35. Complete metric space definition
  36. Continuous map of topological spaces definition
  37. Completeness of the space of bounded continuous functions statement

Where this leads

15 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Uniform_norm, Properties
  2. Web https://en.wikipedia.org/wiki/Complete_metric_space, Examples
About this page
Tag09GRCZVR5E
URIm2://09GRCZVR5E
Kindstatement, theorem
WritingW0 stub
MSC54C35
StatusThis copy of the site has no build data. Run m2 status 09GRCZVR5E.
UsesComplete metric space, Continuous map of topological spaces, Uniform metric on bounded functions

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