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Topologies Extending Valuations

Published online by Cambridge University Press:  20 November 2018

Thomas Rigo  and
Seth Warner
Thomas Rigo
Affiliation:
Indiana University-Pur due University at Indianapolis, Indianapolis, Indiana 46205
Seth Warner
Affiliation:
Duke University, Durham, North Carolina 27706
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Let K be a field complete for a proper valuation (absolute value) v. It is classic that a finite-dimensional K-vector space E admits a unique Hausdorff topology making it a topological K-vector space, and that that topology is the “cartesian product topology” in the sense that for any basis c1 …, cn of E, is a topological isomorphism from Kn to E [1, Chap. I, § 2, no. 3; 2, Chap. VI, § 5, no. 2]. It follows readily that any multilinear mapping from Em to a Hausdorff topological K-vector space is continuous. In particular, any multiplication on E making it a K-algebra is continuous in both variables. If for some such multiplication E is a field extension of K, then by valuation theory the unique Hausdorff topology of E is given by a valuation (absolute value) extending v.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 1 , 01 February 1978 , pp. 164 - 169
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Bourbaki, N., Espaces vectoriels topologiques, Ch. I-11, 2d ed. (Hermann, Paris, 1966).Google Scholar
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3. Applications of model theory to algebra, analysis, and probability, ed. W. A. J. Luxemburg New York, (1969).Google Scholar