Let $G$ be a compact locally $\Bbb{Q}_p$ analytic group. Let $E$ be a finite extension of $\Bbb{Q}_p$ with ring of integers $O$. Let $M$ be a $O[G]$ module.
I was reading an article which says : Let $M$ be a p-adically complete and separated as a $O$ module. What does it mean to be separated. Also how to show that the image of $M$ in $E \otimes_{O} M$ by the natural map becomes an $O$ lattice whose gauge is the required complete norm on $E \otimes_{O} M$ . In this case $E \otimes_{O} M$ is equipped with a natural $E$- banach space structure.
Separated means that the intersection $\cap p^{n} M$ is zero. Where the notation $p^{n}M$ means the submodule of M of the form $(p^{n}) \dot M$. As a $\mathcal{O}$ module the ring $\mathcal{O}_{E}$ is free of finite rank and hence $\mathcal{O}_{E} \otimes M$ is a finite direct sum of $M$. The $\mathcal{O}_{E}$ module structure can be seen explicitly by choosing a basis of $\mathcal{O}_{E}$ over $\mathcal{O}$ (which can be chosen to be cyclic by normal basis theorem). The $E$ module structure comes by inverting the uniformizer.
To answer you comment: Note that $\mathcal{O}$ is a $k$ banach space (where $k$) is the residue field of E with the $p-adic$ norm. If M is finitely generated then there is a presentation of $M$ as a quotient of a free module $\mathcal{O}^{n}$. $\mathcal{O}^{n}$ is a Banach space and it is a fact that any ideal is closed in the $p-adic$ topology so $M$ has a well defined Banach space structure.
Because $\mathcal{O}$ is complete hence $\mathcal{O}^{n}$ and $M$ are complete.