Leigh.Samphier/Sandbox/Inclusion Mapping on Normed Division Subring is Distance Preserving Monomorphism
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Theorem
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\struct {S, \norm {\, \cdot \,}_S}$ be a normed division subring of $\struct {R, \norm {\, \cdot \,}}$.
Then the inclusion mapping $i : S \to R$ is a distance-preserving ring monomorphism.
Proof
From Inclusion Mapping on Subring is Monomorphism, $i$ is a ring monomorphism.
Let $d$ be the metric induced by the norm $\norm {\, \cdot \,}$.
Let $d_S$ be the metric induced by the norm $\norm {\, \cdot \,}_S$.
Thus:
| \(\, \displaystyle \forall x, y \in S : \, \) | \(\displaystyle \map d {\map i x, \map i y}\) | \(=\) | \(\displaystyle \map d {x, y}\) | Definition of Inclusion Mapping | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \norm {x - y}\) | Definition of Metric Induced by Norm on Division Ring | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \norm {x - y}_S\) | Definition of Norm on Subring | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \map {d_S} {x, y}\) | Definition of Metric Induced by Norm on Division Ring |
It follows that $i$ is distance-preserving by definition.
$\blacksquare$
