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
…
…
symmetric monoidal (∞,1)-category of spectra
A normed field $K$ 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 $f,g \in K$ we have the equality ${\vert f \cdot g\vert} = {\vert f\vert} \cdot {\vert g\vert}$ instead of just the inequality ${\vert f \cdot g\vert} \leq {\vert f\vert} \cdot {\vert g\vert}$) 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 $1$ (except that the norm of $0$ is $0$) 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 $\mathbb{R}$ of real numbers and the field $\mathbb{C}$ of complex numbers, with their usual absolute value as the norm, are complete archimedian normed fields.
For each prime number $p$, the field $\mathbb{Q}_p$ of $p$-adic numbers is a complete non-archimedean normed field with respect to the p-adic valuation.
The field $\mathbb{Q}$ of rational numbers, with any of the norms in the two previous entries, is an incomplete normed field whose completion is $\mathbb{R}$ or $\mathbb{Q}_p$.
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 $\mathbb{C}_p$ arise this way from the p-adic rational numbers $\mathbb{Q}_p$.
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 $p$-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.