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A short elementary proof of the Bishop–Stone–Weierstrass theorem

  • T. J. Ransford (a1)

Fix the following notation. Let X be a compact Hausdorff space, and denote by C(X) the vector space of continuous complex-valued functions on X, equipped with the uniform norm ∥·∥x. Let A be a unital subalgebra of C(X). A non-empty subset S of X is said to be A-antisymmetric if whenever hA and h is real-valued on S then h is constant on S.

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[1]Bishop, E.. A generalization of the Stone-Weierstrass theorem. Pacific J. Math. 11 (1961), 777783.
[2] de Branges, L.. The Stone-Weierstrass theorem. Proc. Amer. Math. Soc. 10 (1959), 822824.
[3] Brosowski, B. & Deutsch, F.. An elementary proof of the Stone-Weierstrass theorem. Proc. Amer. Math. Soc. 81 (1981), 8992.
[4] Burckel, R. B.. Bishop' Stone-Weierstrass theorem. Amer. Math. Monthly 91 (1984), 2232.
[5] Glicksberg, I.. Measures orthgonal to algebras and sets of antisymmetry. Trans. Amer. Math. Soc. 105 (1962), 415435.
[6] Machado, S.. On Bishop's generalization of the Stone-Weierstrass theorem. Indag. Math. 39 (1977), 218224.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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