Topologies Extending Valuations
Canadian journal of mathematics, Tome 30 (1978) no. 1, pp. 164-169
Voir la notice de l'article provenant de la source Cambridge University Press
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 K n to E [1, Chap. I, § 2, no. 3; 2, Chap. VI, § 5, no. 2]. It follows readily that any multilinear mapping from E m 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.
Rigo, Thomas; Warner, Seth. Topologies Extending Valuations. Canadian journal of mathematics, Tome 30 (1978) no. 1, pp. 164-169. doi: 10.4153/CJM-1978-015-6
@article{10_4153_CJM_1978_015_6,
author = {Rigo, Thomas and Warner, Seth},
title = {Topologies {Extending} {Valuations}},
journal = {Canadian journal of mathematics},
pages = {164--169},
year = {1978},
volume = {30},
number = {1},
doi = {10.4153/CJM-1978-015-6},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1978-015-6/}
}
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