Category:Definitions/Quotient Topological Vector Spaces

This category contains definitions related to Quotient Topological Vector Spaces.
Related results can be found in Category:Quotient Topological Vector Spaces.

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