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

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.41$: Theorem