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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A Stone-Weierstrass theorem without closure under suprema
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by R. Preston McAfee and Philip J. Reny
Proc. Amer. Math. Soc. 114 (1992), 61-67
DOI: https://doi.org/10.1090/S0002-9939-1992-1091186-2
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Abstract:

For a compact metric space $X$, consider a linear subspace $A$ of $C\left ( X \right )$ containing the constant functions. One version of the Stone-Weierstrass Theorem states that, if $A$ separates points, then the closure of $A$ under both minima and maxima is dense in $C\left ( X \right )$. By the Hahn-Banach Theorem, if $A$ separates probability measures, $A$ is dense in $C\left ( X \right )$. It is shown that if $A$ separates points from probability measures, then the closure of $A$ under minima is dense in $C\left ( X \right )$. This theorem has applications in economic theory.
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Bibliographic Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 114 (1992), 61-67
  • MSC: Primary 41A65; Secondary 41A10, 46E25, 54C40
  • DOI: https://doi.org/10.1090/S0002-9939-1992-1091186-2
  • MathSciNet review: 1091186