Math 2.0

Operator theory Theorem

Linear maps on finite-dimensional spaces are continuous

Builds on Bounded linear maps are continuousEquivalence of norms in finite dimensionContinuous linear map

Statement

Let VV be a finite-dimensional normed space and let WW be a normed space, both over ℝ\mathbb{R} or over ℂ\mathbb{C}. Then each linear map T ⁣:V→WT \colon V \to W is continuous.

Remarks

Proof: choose a basis e1,…,ene_1, \ldots, e_n of VV. For x=∑iaieix = \sum_i a_i e_i, the triangle inequality gives ∥Tx∥W≤(∑i∥Tei∥W)max⁡i∣ai∣\|Tx\|_W \le \left( \sum_i \|T e_i\|_W \right) \max_i |a_i|. By equivalence of norms in finite dimension, max⁡i∣ai∣≤c∥x∥V\max_i |a_i| \le c \|x\|_V for some cc. Thus TT is bounded, and so continuous. The finite dimension of WW is not needed, only that of VV. For example, each matrix gives a continuous map ℝn→ℝm\mathbb{R}^n \to \mathbb{R}^m.

Reading path

To understand this page from the ground up, read these 56 pages in order. The foundations come first.

  1. Element definition
  2. Euclidean metric example
  3. Proposition concept
  4. Real numbers object
  5. Set bounded above definition
  6. p-metric on R^n example
  7. Set definition
  8. Conjunction definition
  9. Disjunction definition
  10. Implication definition
  11. Predicate definition
  12. Supremum definition
  13. p-norm on R^n example
  14. Empty set object
  15. Family of sets definition
  16. Ordered pair definition
  17. Intersection definition
  18. Union definition
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. Cartesian product definition
  22. Subset definition
  23. Union of a family of sets definition
  24. Uniqueness quantifier definition
  25. Binary relation definition
  26. Universal set definition
  27. Topological space definition
  28. Complement definition
  29. Function definition
  30. Subspace topology definition
  31. Preimage of a set definition
  32. Metric space definition
  33. Closed ball definition
  34. Convergence of a sequence in a metric space definition
  35. Lipschitz map definition
  36. Open ball definition
  37. Bounded set definition
  38. Epsilon-delta uniform continuity statement
  39. Metric topology definition
  40. Lipschitz maps are uniformly continuous statement
  41. Open set definition
  42. Closed set definition
  43. Continuous map of topological spaces definition
  44. Equivalent metrics definition
  45. Open cover definition
  46. Equivalent norms definition
  47. Compact space definition
  48. Continuous linear map definition
  49. Compact set definition
  50. Heine–Borel theorem statement
  51. Bounded linear maps are continuous statement
  52. Compactness via open covers statement
  53. Continuous image of a compact set is compact statement
  54. Continuous real function on a compact set attains its bounds statement
  55. Equivalence of norms in finite dimension statement
  56. Linear maps on finite-dimensional spaces are continuous statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Bounded_operator, Examples
About this page
Tag0C13E623WD
URIm2://0C13E623WD
Kindstatement, theorem
WritingW0 stub
MSC47B01
StatusThis copy of the site has no build data. Run m2 status 0C13E623WD.
UsesBounded linear maps are continuous, Equivalence of norms in finite dimension, Continuous linear map

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