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1 results

  • 7 - The Stone–Weierstrass Theorem

  • Adam Bobrowski, Politechnika Lubelska, Poland
  • Book:
    Functional Analysis Revisited
    Published online:
    31 October 2024
    Print publication:
    04 July 2024, pp 70-77

  • 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.