Open Mathematics Courses

Function algebras and approximation

A complete lesson on uniform approximation by algebras of functions on locally compact Hausdorff spaces, without countability or metrizability assumptions.

Includes the real and complex Stone–Weierstrass theorems, compactification, Urysohn’s lemma, lattice approximation, Bernstein polynomials, Korovkin’s theorem, the disc algebra, integer coefficients, and four exercises with solutions.

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Contents

  1. 1. Conventions
  2. 2. Spaces of bounded and continuous functions
  3. 3. The one-point compactification
  4. 4. Compact and locally compact spaces
  5. 5. Urysohn's lemma and metrizability
  6. 6. The polynomial step: absolute values without constant terms
  7. 7. Interpolation at two points
  8. 8. Stone's lattice lemma
  9. 9. The real Stone–Weierstrass theorem
  10. 10. The complex Stone–Weierstrass theorem
  11. 11. Worked examples
  12. 12. The role of each hypothesis
  13. 13. The closure of an arbitrary self-adjoint subalgebra
  14. 14. The Weierstrass theorem: Bernstein polynomials and Korovkin's theorem
  15. 15. Further density theorems on compact spaces
  16. 16. Compact sets, extension of functions, and bounded functions
  17. 17. Convex functions, singular functions, and the disc algebra
  18. 18. Integer coefficients and simultaneous approximation of derivatives
  19. Exercises
  20. Where this leads
  21. Results used from other lessons
  22. References

Authorship and prerequisites

Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session. The October revisions correct two points found by an AI reviewer in a separate OpenAI Codex session and add proofs of the background facts that the core courses leave as exercises; they are self-checked by the writing AI. Public domain (CC0).

The lesson gives exact proof locators in the programme’s hosted Basic Analysis I & II by Jiří Lebl and the core Discrete Mathematics by Oscar Levin. The terminology remark about holomorphic functions links to the published Cauchy-theorem lesson.

English · original exposition under CC0 1.0 · historical sources retain their credit and rights.

Attribution and component terms