Metric Induced by Norm is Invariant Metric

Theorem

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

Let $d$ be the metric induced by $\norm {\, \cdot \,}$.

Then $d$ is invariant.

Proof

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

Then, we have:

\(\ds \map d {x + z, y + z}\)

\(=\)

\(\ds \norm {\paren {x + z} - \paren {y + z} }\)

Definition of Metric Induced by Norm

\(\ds \)

\(=\)

\(\ds \norm {x - y + z - z}\)

\(\ds \)

\(=\)

\(\ds \norm {x - y}\)

\(\ds \)

\(=\)

\(\ds \map d {x, y}\)

Definition of Metric Induced by Norm

$\blacksquare$