Math 2.0

Operator theory Definition

Continuous linear map

Builds on Equivalent normsContinuous map of topological spaces

Definition

Let VV and WW be vector spaces over ℝ\mathbb{R} or ℂ\mathbb{C}, with norms ∥⋅∥V\|\cdot\|_V and ∥⋅∥W\|\cdot\|_W. A map T ⁣:V→WT \colon V \to W is a continuous linear map when TT is linear and continuous for the metrics ∥x−y∥V\|x - y\|_V and ∥x−y∥W\|x - y\|_W.

Remarks

A linear map is continuous if and only if it is bounded, that is, ∥Tx∥W≤C∥x∥V\|Tx\|_W \le C \|x\|_V for some constant CC and all xx. This is bounded linear maps are continuous. The smallest such CC is the operator norm of TT. On a finite-dimensional space, each linear map is continuous. The derivative p↦p′p \mapsto p' on polynomials, with the sup norm on [0,1][0, 1], is linear but not continuous: ∥xn∥∞=1\|x^n\|_\infty = 1 but ∥nxn−1∥∞=n\|n x^{n-1}\|_\infty = n.

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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 spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen 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 mapdefinition
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. Open ball definition
  26. Metric topology definition
  27. Open set definition
  28. Continuous map of topological spaces definition
  29. Equivalent metrics definition
  30. Equivalent norms definition
  31. Continuous linear map definition

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Bounded_operator, In normed vector spaces
About this page
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WritingW1 stated
MSC47B01
DefinesContinuous linear map
UsesEquivalent norms, Continuous map of topological spaces

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