Vol. 109, No. 1, 1983

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Norms on F(X)

Jo-Ann Deborah Cohen

Vol. 109 (1983), No. 1, 81–87

It is well known that if ∥⋅⋅∥ is a norm on the field F(X) of rational functions over a field F for which F is bounded, then ∥⋅⋅∥ is equivalent to the supremum of a finite family of absolute values on F(X), each of which is improper on F. Moreover, ∥⋅⋅∥ is equivalent to an absolute value if and only if the completion of F(X) for ∥⋅⋅∥ is a field. We show that the analogous characterization of norms on F(X) for which F is discrete is impossible by constructing for each infinite field F, a norm ∥⋅⋅∥ on F(X) such that F is discrete, X< 1, the completion of F(X) for ∥⋅⋅∥ is a field, but ∥⋅⋅∥ is not equivalent to the supremum of finitely many absolute values.

Mathematical Subject Classification 2000
Primary: 12J05
Received: 19 October 1981
Revised: 4 November 1982
Published: 1 November 1983
Jo-Ann Deborah Cohen