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 if and only if:

$\forall x \in X: x \ne \mathbf 0_X \implies \exists p \in \PP : \map p x \ne 0$

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

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.33$: Definitions