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

$\blacksquare$

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