Definition:Pseudometric Induced by Seminorm

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

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

Let $p$ be a seminorm on $X$.

Define $d_p : X \times X \to \R$ by:


 * $\map {d_p} {x, y} = \map p {x - y}$

for each $x, y \in X$.

We say that $d_p$ is the pseudometric induced by $p$.

Also see

 * Pseudometric Induced by Seminorm is Pseudometric