Math 2.0

Functional analysis Normed and Banach spaces Theorem

Equivalence of norms in finite dimension

Builds on Equivalent normsHeine–Borel theoremContinuous real function on a compact set attains its boundsp-norm on R^n

Statement

Let VV be a finite-dimensional vector space over ℝ\mathbb{R} or ℂ\mathbb{C}. Then any two norms on VV are equivalent.

Remarks

Proof idea: fix a basis and compare a norm ∥⋅∥\|\cdot\| with the sup norm ∥⋅∥∞\|\cdot\|_\infty of the coordinates. The triangle inequality gives ∥x∥≤C∥x∥∞\|x\| \le C \|x\|_\infty, so ∥⋅∥\|\cdot\| is continuous for ∥⋅∥∞\|\cdot\|_\infty. The unit sphere of ∥⋅∥∞\|\cdot\|_\infty is compact by the Heine–Borel theorem, so ∥⋅∥\|\cdot\| has a positive minimum cc on it, by the extreme value theorem. Thus there is only one norm topology on ℝn\mathbb{R}^n, the Euclidean topology. The statement fails in infinite dimensions: each infinite-dimensional normed space has norms that are not equivalent.

Reading path

To understand this page from the ground up, read these 49 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. Open ball definition
  35. Bounded set definition
  36. Metric topology definition
  37. Open set definition
  38. Closed set definition
  39. Continuous map of topological spaces definition
  40. Equivalent metrics definition
  41. Open cover definition
  42. Equivalent norms definition
  43. Compact space definition
  44. Compact set definition
  45. Heine–Borel theorem statement
  46. Compactness via open covers statement
  47. Continuous image of a compact set is compact statement
  48. Continuous real function on a compact set attains its bounds statement
  49. Equivalence of norms in finite dimension statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Norm_(mathematics), Equivalence
About this page
Tag03MC70ZRNR
URIm2://03MC70ZRNR
Kindstatement, theorem
WritingW1 stated
MSC46B20
StatusThis copy of the site has no build data. Run m2 status 03MC70ZRNR.
UsesEquivalent norms, Heine–Borel theorem, Continuous real function on a compact set attains its bounds, p-norm on R^n

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