Math 2.0

Functional analysis Theorem

Compact closed unit ball implies finite dimension

Builds on Riesz's lemmaCompactness and sequential compactness agree in metric spacesFinite-dimensional normed spaces are completeHeine–Borel theorem in finite-dimensional normed spaces

Statement

Let XX be a normed space over ℝ\mathbb{R} or ℂ\mathbb{C}. If the closed unit ball {x∈X:∥x∥≤1}\{ x \in X : \|x\| \le 1 \} is compact, then XX is finite-dimensional.

Remarks

Proof: if XX is infinite-dimensional, build unit vectors x1,x2,…x_1, x_2, \ldots by Riesz's lemma, with xn+1x_{n+1} at distance at least 12\tfrac{1}{2} from the span of x1,…,xnx_1, \ldots, x_n. That span is finite-dimensional, so it is complete and closed. Then ∥xm−xn∥≥12\|x_m - x_n\| \ge \tfrac{1}{2} for m≠nm \neq n, so no subsequence converges, and the ball is not sequentially compact. With the Heine–Borel theorem in finite dimension, we get: a normed space is finite-dimensional if and only if its closed unit ball is compact. Equivalently, a normed space is locally compact if and only if it is finite-dimensional.

Reading path

To understand this page from the ground up, read these 63 pages in order. The foundations come first.

  1. Element definition
  2. Euclidean metric example
  3. Extended real number line object
  4. Proposition concept
  5. Real numbers object
  6. Set bounded above definition
  7. p-metric on R^n example
  8. Set definition
  9. Conjunction definition
  10. Disjunction definition
  11. Implication definition
  12. Predicate definition
  13. Supremum definition
  14. p-norm on R^n example
  15. Empty set object
  16. Family of sets definition
  17. Ordered pair definition
  18. Intersection definition
  19. Union definition
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Cartesian product definition
  23. Subset definition
  24. Union of a family of sets definition
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Universal set definition
  28. Topological space definition
  29. Complement definition
  30. Function definition
  31. Sequential compactness definition
  32. Subspace topology definition
  33. Preimage of a set definition
  34. Metric space definition
  35. Closed ball definition
  36. Convergence of a sequence in a metric space definition
  37. Open ball definition
  38. Diameter of a set definition
  39. Bounded set definition
  40. Metric topology definition
  41. Epsilon characterisation of Cauchy sequences statement
  42. Complete metric space definition
  43. Open set definition
  44. Closed set definition
  45. Continuous map of topological spaces definition
  46. Equivalent metrics definition
  47. Open cover definition
  48. Equivalent norms definition
  49. Compact space definition
  50. Banach space definition
  51. Riesz's lemma statement
  52. Compact set definition
  53. Lebesgue number lemma statement
  54. Heine–Borel theorem statement
  55. Proper metric space definition
  56. Compactness and sequential compactness agree in metric spaces statement
  57. Compactness via open covers statement
  58. Continuous image of a compact set is compact statement
  59. Continuous real function on a compact set attains its bounds statement
  60. Equivalence of norms in finite dimension statement
  61. Finite-dimensional normed spaces are complete statement
  62. Heine–Borel theorem in finite-dimensional normed spaces statement
  63. Compact closed unit ball implies finite dimension statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Riesz%27s_lemma, Characterization of finite dimension
About this page
Tag0VZVV8VME5
URIm2://0VZVV8VME5
Kindstatement, theorem
WritingW0 stub
MSC46B50
StatusThis copy of the site has no build data. Run m2 status 0VZVV8VME5.
UsesRiesz's lemma, Compactness and sequential compactness agree in metric spaces, Finite-dimensional normed spaces are complete, Heine–Borel theorem in finite-dimensional normed spaces

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