Remark

Let $(G_n)_{n \in \mathbf{Z}}$ be a decreasing sequence of subgroups of $G$. It is a decreasing filtration in the sense of Commutative Algebra, Chapter III, § 2, no. 1, Definition 1. For every integer $n$ and all $\alpha$ in the interval $]n-1, n]$ of $\mathbf{R}$, we write $H_\alpha = G_n$, in particular $H_n = G_n$. It is immediate that we thus obtain a real filtration $(H_\alpha)_{\alpha \in \mathbf{R}}$ on $G$; such a filtration will be called an integral filtration. Hence decreasing filtrations in the sense of Commutative Algebra, Chapter III, § 2, can be identified with integral filtrations.

Let $A$ be an algebra; a real filtration $(A_\alpha)$ on the additive group of $A$ is called compatible with the algebra structure if $A_\alpha . A_\beta \subset A_{\alpha + \beta}$ for $\alpha, \beta$ in $\mathbf{R}$ and $K . A_\alpha \subset A_\alpha$ for $\alpha \in \mathbf{R}$. If the filtration is exhaustive, $(A_\alpha)$ is a fundamental system of neighbourhoods of 0 under the topology on $A$ which is compatible with the algebra structure. Let $B$ be a unital algebra; a real filtration $(B_\alpha)$ on the additive group of $B$ is called compatible with the unital algebra structure if it is compatible with the algebra structure and $1 \in B_0$.