Math 2.0

Operator theory Theorem

Bounded linear maps are continuous

Builds on Continuous linear mapLipschitz mapLipschitz maps are uniformly continuous

Statement

Let VV and WW be normed spaces and let T ⁣:V→WT \colon V \to W be linear. Then TT is continuous if and only if TT is bounded, that is, there is a real C≥0C \ge 0 with ∥Tx∥W≤C∥x∥V\|Tx\|_W \le C \|x\|_V for all x∈Vx \in V.

Remarks

If TT is bounded, then ∥Tx−Ty∥W=∥T(x−y)∥W≤C∥x−y∥V\|Tx - Ty\|_W = \|T(x - y)\|_W \le C \|x - y\|_V. Thus TT is Lipschitz, and so uniformly continuous. Conversely, if TT is continuous at 00, there is δ>0\delta > 0 with ∥Tx∥W≤1\|Tx\|_W \le 1 when ∥x∥V≤δ\|x\|_V \le \delta. By linearity, ∥Tx∥W≤∥x∥V/δ\|Tx\|_W \le \|x\|_V / \delta for all xx. Thus, for a linear map, continuity at one point already gives continuity everywhere. The name bounded means that TT maps bounded sets to bounded sets. It does not mean that T(V)T(V) is bounded.

Reading path

To understand this page from the ground up, read these 36 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 levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty 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…definitionLipschitz map, definition, no evidence levelLipschitz mapdefinitionOpen ball, definition, no evidence levelOpen balldefinitionEpsilon-delta uniform continuity, statement, no evidence levelEpsilon-delta uniform…statementMetric topology, definition, no evidence levelMetric topologydefinitionLipschitz maps are uniformly continuous, statement, no evidence levelLipschitz maps are un…statementOpen set, definition, no evidence levelOpen setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionEquivalent metrics, definition, no evidence levelEquivalent metricsdefinitionEquivalent norms, definition, no evidence levelEquivalent normsdefinitionContinuous linear map, definition, no evidence levelContinuous linear mapdefinitionBounded linear maps are continuous, statement, no evidence levelBounded linear maps a…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. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Predicate definition
  9. Empty set object
  10. Family of sets definition
  11. Ordered pair definition
  12. Intersection definition
  13. Union definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Union of a family of sets definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Topological space definition
  22. Function definition
  23. Preimage of a set definition
  24. Metric space definition
  25. Convergence of a sequence in a metric space definition
  26. Lipschitz map definition
  27. Open ball definition
  28. Epsilon-delta uniform continuity statement
  29. Metric topology definition
  30. Lipschitz maps are uniformly continuous statement
  31. Open set definition
  32. Continuous map of topological spaces definition
  33. Equivalent metrics definition
  34. Equivalent norms definition
  35. Continuous linear map definition
  36. Bounded linear maps are continuous statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Bounded_operator, Equivalence of boundedness and continuity
About this page
Tag0F3K4C7XYW
URIm2://0F3K4C7XYW
Kindstatement, theorem
WritingW0 stub
MSC47B01
StatusThis copy of the site has no build data. Run m2 status 0F3K4C7XYW.
UsesContinuous linear map, Lipschitz map, Lipschitz maps are uniformly continuous

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