We assume that $|\raisebox{1.72218pt}{.}|$ is discrete. Then we have the following:

1. (1)

   Every lattice $\mathscr{V}$ of $V$ is a finitely generated $\mathfrak{o}_{k}$-module.

2. (2)

   If we set $\mathscr{V}:=(V,\|\raisebox{1.72218pt}{.}\|)_{\leq 1}$ for a norm of $\|\raisebox{1.72218pt}{.}\|$ of $V$, then $\|\raisebox{1.72218pt}{.}\|\leq\|\raisebox{1.72218pt}{.}\|_{\mathscr{V}}\leq|\varpi|^{-1}\|\raisebox{1.72218pt}{.}\|$.