## 2.1 Variants of normed groups

This subsection continues some of the theory of (pseudo)-normed groups, started in Subsection 1.2.

A \(p\)-Banach space \(V\) is a complete topological \(\mathbb R\)-vector space whose topology is induced by a \(p\)-norm; that is, a norm satisfying \(\| rv\| = |r|^p\| v\| \).

A \(p\)-Banach \(V\) is (up to non-canonical choice) an \(r\)-Banach \(\mathbb Z[T^{\pm 1}]\)-module, where \(r=2^{-p}\). (See Definition 1.2.1 for the definition of \(r\)-normed \(\mathbb Z[T^{\pm 1}]\)-modules.)

Obvious.

We will consider the following categories:

\(\text{CHPNG}\) the category of CH-filtered pseudo-normed groups with bounded morphisms.

\(\text{CHPNG}_1\) the category of exhaustive CH-filtered pseudo-normed groups with strict morphisms.

\(\text{ProfinPNG}\) the category of profinitely filtered pseudo-normed groups with bounded morphisms.

\(\text{ProfinPNG}_1\) the category of exhaustive profinitely filtered pseudo-normed groups with strict morphisms.

\(\text{ProfinPNGTinv}_1^{r'}\) the category of exhaustive profinitely filtered pseudo-normed groups with \(r'\)-action of \(T^{-1}\) and strict morphisms.

Consider an inverse system \((X_i)_i\) of compact-Hausdorffly-filtered-pseudonormed abelian groups where all transition maps \(X_i\to X_j\) send \(X_{i,\leq c}\) to \(X_{j,\leq c}\). Then

is compact Hausdorff, and

is naturally a compact-Hausdorffly-filtered pseudonormed abelian group which is the limit of \((X_i)_i\) in the strict category structure.

One can define negation and addition on \(X\) as continuous maps \(-: X_{\leq c}\to X_{\leq c}\) and \(+: X_{\leq c}\times X_{\leq c'}\to X_{\leq c+c'}\), and these pass to the colimit. It should then be straightforward to check the axioms.