A non-Archimedean Stone-Banach theorem
Edward Beckenstein and Lawrence Narici
Proc. Amer. Math. Soc. 100 (1987), 242-246
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Abstract: If the spaces and of continuous functions on and are linearly isometric, then and are homeomorphic. By the classical Stone-Banach theorem the only linear isometries of onto are of the form , where is a homeomorphism of onto and is of magnitude 1 for all in . What happens if is replaced by a field with a valuation? In brief, the result fails. We discuss "how" by way of developing a necessary and sufficient condition for the theorem to hold, along with some examples to illustrate the point.
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