\(\newcommand{\ff}{\mathbb{F}} \DeclareMathOperator{\ord}{ord} \newcommand{\qq}{\mathbb{Q}} \newcommand{\rr}{\mathbb{R}} \newcommand{\rnneg}{\rr_{\geq 0}} \newcommand{\rpos}{\rr_{> 0}} \newcommand{\zz}{\mathbb{Z}} \newcommand{\znneg}{\zz_{\geq 0}} \)Given a normed (abelian) group \(A,\) the following are some important subsets of \(\prod_{i=1}^\infty A\): It is clear that \(A^{(\infty)} \subseteq c(A) \subseteq b(A).\) If \(A\) is a ring/algebra over a field/module over a ring, each of these sets inherits the structure from \(A.\)… Continue reading Notes (part 2): S. Bosch, U Güntzer and R. Remmert, Non-Archimedean Analysis
Tag: Non-Archimedean
Notes (part 1): S. Bosch, U Güntzer and R. Remmert, Non-Archimedean Analysis
Semi-normed and normed groups \(\newcommand{\mmm}{\mathfrak{m}} \newcommand{\qqq}{\mathfrak{q}} \newcommand{\rr}{\mathbb{R}} \newcommand{\rnneg}{\rr_{\geq 0}} \newcommand{\rpos}{\rr_{> 0}} \newcommand{\zz}{\mathbb{Z}} \newcommand{\znneg}{\zz_{\geq 0}} \)All groups are going to abelian with the group operation denoted by “+”. A filtration, a generalization of valuation, is a function \(\nu\) from a group \(G\) to \(\rr \cup \{\infty\}\) such that Filtrations are in one-to-one correspondence with ultrametric functions… Continue reading Notes (part 1): S. Bosch, U Güntzer and R. Remmert, Non-Archimedean Analysis