Quotient Metric on Vector Space is Well-Defined

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.

Then the mapping $d_N : X/N \times X/N \to \hointr 0 \infty$ defined by:

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

for each $\map \pi x, \map \pi y \in X/N$, is well-defined.

Proof

Let $x, y \in X$.

Then $\map d {x - y, z} \ge 0$ for all $z \in N$, and so:

$\ds \inf_{z \mathop \in N} \map d {x - y, z}$ exists as a real number.

Let $x', y' \in X$ be such that $\map \pi x = \map \pi {x'}$ and $\map \pi y = \map \pi {y'}$.

We now need to show that if $x', y' \in X$ are such that:

$\map \pi x = \map \pi {x'}$

and:

$\map \pi y = \map \pi {y'}$

then:

$\ds \inf_{z \mathop \in N} \map d {x - y, z} = \inf_{z \mathop \in N} \map d {x' - y', z}$

From Quotient Mapping is Linear Transformation, we have $\map \pi {x' - x} = {\mathbf 0}_{X/N}$ and $\map \pi {y' - y} = {\mathbf 0}_{X/N}$.

From Kernel of Quotient Mapping, we obtain $x' \in x + N$ and $y' \in y + N$.

Then there exists $z_1, z_2 \in N$ such that $x' = x + z_1$ and $y' = y + z_2$.

Then we have:

\(\ds \inf_{z \mathop \in N} \map d {x' - y', z}\)

\(=\)

\(\ds \inf_{z \mathop \in N} \map d {x + z_1 - \paren {y + z_2}, z}\)

\(\ds \)

\(=\)

\(\ds \inf_{z \mathop \in N} \map d {\paren {x - y} + \paren {z_1 - z_2}, z}\)

\(\ds \)

\(=\)

\(\ds \inf_{z \mathop \in N} \map d {x - y, z - \paren {z_1 - z_2} }\)

Definition of Invariant Metric on Vector Space

\(\ds \)

\(=\)

\(\ds \inf_{z' \mathop \in N - \paren {z_1 - z_2} } \map d {x - y, z'}\)

\(\ds \)

\(=\)

\(\ds \inf_{z' \mathop \in N} \map d {x - y, z'}\)

Definition of Vector Subspace

as required.

$\blacksquare$