Vol. 27, No. 2, 1968

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The Stone-Weierstrass theorem for valuable fields

Paul Robert Chernoff, Richard Anthony Rasala and William Charles Waterhouse

Vol. 27 (1968), No. 2, 233–240

Let F be a topological field, which means in particular Hausdorff and commutative. The “Stone-Weierstrass” theorem for F would state that if X is a compact space and 𝒜 an algebra of continuous functions from X to F which contains the constant functions and separates points, then 𝒜 is uniformly dense in the algebra of all continuous functions from X to F. This is true for the reals but false for the complexes, and is commonly regarded as a special property of the reals. In fact, however, it is the complex field which is exceptional: the Stone-Weierstrass theorem and its characteristic corollaries hold for all valuable fields other than the complex numbers.

Mathematical Subject Classification
Primary: 12.70
Received: 5 October 1967
Published: 1 November 1968
Paul Robert Chernoff
Richard Anthony Rasala
William Charles Waterhouse