Definition:Separating Family of Seminorms on Vector Space

Definition
Let $\GF \in \set {\R, \C}$.

Let $X$ be a vector space over $\GF$.

Let $\PP$ be a set of seminorms on $X$.

We say that $\PP$ is separating :
 * $\forall x \in X: x \ne \mathbf 0_X \implies \exists p \in \PP : \map p x \ne 0$

where $\mathbf 0_X$ denotes the zero vector in $X$.