analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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symmetric monoidal (∞,1)-category of spectra
A normed field is a normed ring whose underlying ring is a field.
If the product preserves the norm strictly (so that one has a multiplicative norm or absolute value in that for all we have the equality instead of just the inequality ) then one speaks of a valued field (e.g. Berkovich 09, def. 1.1.1).
If the underlying normed group is a complete topological space then one speaks of a complete normed field.
Every field carries the trivial norm (which is non-archimedean), whose value is always (except that the norm of is ) and is complete with respect to this norm. (In constructive mathematics, either the field must be a discrete field or the norm must be allowed to take values in the lower real numbers.)
The field of real numbers and the field of complex numbers, with their usual absolute value as the norm, are complete archimedian normed fields.
For each prime number , the field of -adic numbers is a complete non-archimedean normed field with respect to the p-adic valuation.
The field of rational numbers, with any of the norms in the two previous entries, is an incomplete normed field whose completion is or .
The norm of a non-archimedean field extends uniquely to its algebraic closure and the completion of that with respect to this norm is still algebraically closed (Bosch-Guntzer-Remmert 84, prop. 3.4.1.3, Berkovich 09, fact 1.1.4).
For example the p-adic complex numbers arise this way from the p-adic rational numbers .
Leonard Tornheim, Normed fields over the real and complex fields, Michigan Math. J. Volume 1, Issue 1 (1952), 61-68. (Euclid)
S. Bosch, U. Guntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften, vol. 261, Springer-Verlag, Berlin, 1984.
Vladimir Berkovich, Non-archimedean analytic spaces, lectures at the Advanced School on -adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)
Last revised on July 13, 2014 at 07:58:57. See the history of this page for a list of all contributions to it.