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

Sources

• 2018: Henri Bourlès: Fundamentals of Advanced Mathematics 2 ... (previous) ... (next): $3.3.2$: Seminorms