Quotient Metric on Vector Space is Invariant Pseudometric

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Theorem

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.

Let $d_N$ be the quotient metric on $X/N$ induced by $d$.


Then $d_N$ is an invariant pseudometric.


Proof

Proof of Metric Space Axiom $(\text M 1)$

Let $x, y \in X$.

Then, we have:

$\ds \map {d_N} {\map \pi x, \map \pi x} = \inf_{z \mathop \in N} \map d {x - x, z} = \inf_{z \mathop \in N} \map d { {\mathbf 0}_X, z}$

Since $N$ is a vector subspace, we have ${\mathbf 0}_X \in N$.

From Metric Space Axiom $(\text M 1)$, we have $\map d { {\mathbf 0}_X, {\mathbf 0}_X} = 0$, and so:

$\ds \inf_{z \mathop \in N} \map d { {\mathbf 0}_X, z} \le 0$

Since we also have:

$\ds \inf_{z \mathop \in N} \map d { {\mathbf 0}_X, z} \ge 0$

we obtain:

$\ds \inf_{z \mathop \in N} \map d { {\mathbf 0}_X, z} = 0$

Hence we have proved Metric Space Axiom $(\text M 1)$ for $d_N$.

$\Box$

Proof of Metric Space Axiom $(\text M 2)$: Triangle Inequality

Let $x, y, z \in X$.

Applying Metric Space Axiom $(\text M 2)$: Triangle Inequality to $d$, we have:

$\map d {x, z + n} \le \map d {x, y + n'} + \map d {y + n', z + n}$

for each $n, n' \in N$.

That is, using the translation invariance of $d$:

$\map d {x - z, n} \le \map d {x - y, n'} + \map d {y - z, n - n'}$

for each $n, n' \in N$.

Taking the infimum over $n \in N$, we have:

\(\ds \map {d_N} {\map \pi x, \map \pi z}\) \(=\) \(\ds \inf_{n \mathop \in N} \map d {x - z, n}\) Definition of Invariant Metric on Vector Space
\(\ds \) \(\le\) \(\ds \inf_{n \mathop \in N} \paren {\map d {x - y, n'} + \map d {y - z, n - n'} }\) Infimum preserves Inequalities
\(\ds \) \(=\) \(\ds \map d {x - y, n'} + \inf_{n \mathop \in N} \map d {y - z, n - n'}\) Infimum Plus Constant
\(\ds \) \(=\) \(\ds \map d {x - y, n'} + \inf_{n^\ast \mathop \in N - n'} \map d {y - z, n^\ast}\)
\(\ds \) \(=\) \(\ds \map d {x - y, n'} + \inf_{n^\ast \mathop \in N} \map d {y - z, n^\ast}\) Definition of Vector Subspace
\(\ds \) \(=\) \(\ds \map d {x - y, n'} + \map {d_N} {\map \pi y, \map \pi z}\) Definition of Quotient Metric on Vector Space

Taking the infimum over $n' \in N$, using Infimum preserves Inequalities and Infimum Plus Constant, we have:

$\map {d_N} {\map \pi x, \map \pi z} \le \map {d_N} {\map \pi x, \map \pi y} + \map {d_N} {\map \pi y, \map \pi z}$

Hence we have proved Metric Space Axiom $(\text M 2)$: Triangle Inequality for $d_N$.

$\Box$

Proof of Metric Space Axiom $(\text M 3)$

Let $x, y \in X$.

Then we have:

\(\ds \map {d_N} {\map \pi x, \map \pi y}\) \(=\) \(\ds \inf_{z \mathop \in N} \map d {x - y, z}\)
\(\ds \) \(=\) \(\ds \inf_{z \mathop \in N} \map d {y - x, -z}\) Symmetry of Invariant Metric on Vector Space
\(\ds \) \(=\) \(\ds \inf_{z' \mathop \in -N} \map d {y - x, z'}\)
\(\ds \) \(=\) \(\ds \inf_{z \mathop \in N} \map d {y - x, z}\) Definition of Vector Subspace
\(\ds \) \(=\) \(\ds \map {d_N} {\map \pi y, \map \pi x}\) Definition of Quotient Metric on Vector Space

Hence we have proved Metric Space Axiom $(\text M 3)$ for $d_N$.

$\Box$

Proof of translation invariance

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

We have:

\(\ds \map {d_N} {\map \pi x + \map \pi z, \map \pi y + \map \pi z}\) \(=\) \(\ds \map {d_N} {\map \pi {x + z}, \map \pi {y + z} }\) Quotient Mapping is Linear Transformation
\(\ds \) \(=\) \(\ds \inf_{n \mathop \in N} \map d {\paren {x + z} - \paren {y + z}, n}\)
\(\ds \) \(=\) \(\ds \inf_{n \mathop \in N} \map d {x - y, n}\)
\(\ds \) \(=\) \(\ds \map {d_N} {\map \pi x, \map \pi y}\)

So $d_N$ is an invariant pseudometric.

$\blacksquare$

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