Definition:Quotient Metric on Vector Space

Definition
Let $K$ be a field.

Let $X$ be a vector space over $K$.

Let $d$ be an invariant metric on $X$.

Let $N$ be a vector subspace of $X$.

Let $X/N$ be the quotient vector space of $X$ modulo $N$.

Let $\pi : X \to X/N$ be the quotient mapping.

We define the quotient metric on $X/N$ induced by $d$ by:
 * $\ds \map {d_N} {\map \pi x, \map \pi y} = \inf_{z \mathop \in N} \map d {x - y, z}$

for each $\map \pi x, \map \pi y \in X/N$.

Also see

 * Quotient Metric on Vector Space is Well-Defined
 * Quotient Metric on Vector Space is Invariant Pseudometric
 * Quotient Metric on Vector Space is Invariant Metric iff Vector Subspace is Closed
 * Quotient Metric induced by Norm is Quotient Norm