Math 2.0

Operator theory Definition

Operator norm

Builds on Continuous linear mapBounded linear maps are continuousSupremum

Definition

Let VV and WW be normed spaces and let T ⁣:V→WT \colon V \to W be a continuous linear map. The operator norm of TT is ∥T∥=sup⁡{∥Tx∥W:x∈V, ∥x∥V≤1}.\|T\| = \sup \{ \|Tx\|_W : x \in V, \ \|x\|_V \le 1 \}.

Remarks

The supremum is finite because TT is bounded. If V≠{0}V \neq \{0\}, then ∥T∥=sup⁡x≠0∥Tx∥W/∥x∥V\|T\| = \sup_{x \neq 0} \|Tx\|_W / \|x\|_V. It is the smallest C≥0C \ge 0 with ∥Tx∥W≤C∥x∥V\|Tx\|_W \le C \|x\|_V for all x∈Vx \in V. The operator norm is a norm on the space L(V,W)L(V, W) of continuous linear maps. It satisfies ∥ST∥≤∥S∥ ∥T∥\|ST\| \le \|S\| \, \|T\|. If WW is a Banach space, so is L(V,W)L(V, W). For a diagonal matrix on ℝn\mathbb{R}^n with the Euclidean norm, the operator norm is the largest absolute value of a diagonal entry.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSet, 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…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…statementOperator norm, definition, no evidence levelOperator normdefinition
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  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set bounded above definition
  5. Set definition
  6. Conjunction definition
  7. Disjunction definition
  8. Implication definition
  9. Predicate definition
  10. Supremum definition
  11. Empty set object
  12. Family of sets definition
  13. Ordered pair definition
  14. Intersection definition
  15. Union definition
  16. Existential quantifier definition
  17. Universal quantifier definition
  18. Cartesian product definition
  19. Subset definition
  20. Union of a family of sets definition
  21. Uniqueness quantifier definition
  22. Binary relation definition
  23. Topological space definition
  24. Function definition
  25. Preimage of a set definition
  26. Metric space definition
  27. Convergence of a sequence in a metric space definition
  28. Lipschitz map definition
  29. Open ball definition
  30. Epsilon-delta uniform continuity statement
  31. Metric topology definition
  32. Lipschitz maps are uniformly continuous statement
  33. Open set definition
  34. Continuous map of topological spaces definition
  35. Equivalent metrics definition
  36. Equivalent norms definition
  37. Continuous linear map definition
  38. Bounded linear maps are continuous statement
  39. Operator norm definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Operator_norm, Introduction and definition
  2. Web https://en.wikipedia.org/wiki/Operator_norm, Equivalent definitions
About this page
Tag0ZTEH055YF
URIm2://0ZTEH055YF
Kinddefinition
WritingW1 stated
MSC47A30
DefinesOperator norm
UsesContinuous linear map, Bounded linear maps are continuous, Supremum

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