Math 2.0

Functional analysis Theorem

Heine–Borel theorem in finite-dimensional normed spaces

Builds on Heine–Borel theoremEquivalence of norms in finite dimensionProper metric spacep-norm on R^n

Statement

Let VV be a finite-dimensional vector space over ℝ\mathbb{R} or ℂ\mathbb{C}, with a norm ∥⋅∥\|\cdot\| and the metric d(x,y)=∥x−y∥d(x, y) = \|x - y\|. Then a subset K⊆VK \subseteq V is compact if and only if KK is closed and bounded.

Remarks

Proof: choose a basis. By equivalence of norms in finite dimension, the norm is equivalent to the Euclidean norm of the coordinates. Equivalent norms have the same closed sets, bounded sets and compact sets. Then use the Heine–Borel theorem in ℝn\mathbb{R}^n or ℂn=ℝ2n\mathbb{C}^n = \mathbb{R}^{2n}. In other words, each finite-dimensional normed space is a proper metric space. The converse also holds: if the closed unit ball is compact, the space is finite-dimensional.

Reading path

To understand this page from the ground up, read these 51 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. Proper metric space definition
  46. Heine–Borel theorem statement
  47. Compactness via open covers statement
  48. Continuous image of a compact set is compact statement
  49. Continuous real function on a compact set attains its bounds statement
  50. Equivalence of norms in finite dimension statement
  51. Heine–Borel theorem in finite-dimensional normed spaces statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem, In the theory of topological vector spaces
  2. Web https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem, In the theory of metric spaces
About this page
Tag0PHT6NBYPB
URIm2://0PHT6NBYPB
Kindstatement, theorem
WritingW0 stub
MSC46B20
StatusThis copy of the site has no build data. Run m2 status 0PHT6NBYPB.
UsesHeine–Borel theorem, Equivalence of norms in finite dimension, Proper metric space, p-norm on R^n

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