Book contents
- Frontmatter
- Contents
- Introduction
- 1 Complete Metric Spaces
- 2 Banach’s Principle
- 3 Picard’s Theorem
- 4 Banach Spaces
- 5 Renewal Equation in the McKendrick–von Foerster Model
- 6 Riemann Integral for Vector-Valued Functions
- 7 The Stone–Weierstrass Theorem
- 8 Norms Do Differ
- 9 Hilbert Spaces
- 10 Complete Orthonormal Sequences
- 11 Heat Equation
- 12 Completeness of the Space of Operators
- 13 Working in ℒ(𝕏)
- 14 The Banach–Steinhaus Theorem and Strong Convergence
- 15 We Go Deeper, DeeperWe Go (into the Structure of Complete Spaces)
- 16 Semigroups of Operators
- Appendix Two Consequences of the Hahn–Banach Theorem
- References
- Index
7 - The Stone–Weierstrass Theorem
Published online by Cambridge University Press: 31 October 2024
Book contents
- Frontmatter
- Contents
- Introduction
- 1 Complete Metric Spaces
- 2 Banach’s Principle
- 3 Picard’s Theorem
- 4 Banach Spaces
- 5 Renewal Equation in the McKendrick–von Foerster Model
- 6 Riemann Integral for Vector-Valued Functions
- 7 The Stone–Weierstrass Theorem
- 8 Norms Do Differ
- 9 Hilbert Spaces
- 10 Complete Orthonormal Sequences
- 11 Heat Equation
- 12 Completeness of the Space of Operators
- 13 Working in ℒ(𝕏)
- 14 The Banach–Steinhaus Theorem and Strong Convergence
- 15 We Go Deeper, DeeperWe Go (into the Structure of Complete Spaces)
- 16 Semigroups of Operators
- Appendix Two Consequences of the Hahn–Banach Theorem
- References
- Index
Summary
The Weiestrass theorem says that any continuous function on a finite closed interval can be uniformly approximated, with any required accuracy, by polynomials. The Stone–Weierstrass theorem extends this result to an abstract setting, where the interval is replaced by a compact topological space, and the role of polynomials is played by a class of functions that enjoy certain properties mimicking those of polynomials. There are scores of proofs of the latter result; the one presented in this little book could not fail to stress the importance of completeness of the space of continuous functions.
Keywords
approximation of continuous functions by polynomialscontinuous functions on abstractcompact topological spacesseparation of pointsdense sub-algebras of the space of continuous functionsexistence of roots in this spacelatticesKakutani–Krein and Stone–Weierstrass theoremsHausdorff moment problemtrigonometric polynomials
- Type
- Chapter
- Information
- Functional Analysis RevisitedAn Essay on Completeness, pp. 70 - 77Publisher: Cambridge University PressPrint publication year: 2024