Division Subring of Normed Division Ring

Theorem

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

Let $S$ be a division subring of $R$.

Then:

$\struct {S, \norm {\, \cdot \,}_S}$ is a normed division subring of $\struct {R, \norm {\, \cdot \,} }$

where $\norm {\, \cdot \,}_S$ is the norm $\norm {\,\cdot\,}$ restricted to $S$.

Proof

Norm Axiom $\text N 1$: Positive Definiteness

\(\ds \forall x \in S: \, \)

\(\ds \norm x_S\)

\(=\)

\(\ds 0\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \norm x\)

\(=\)

\(\ds 0\)

Definition of $\norm x_S$

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=\)

\(\ds 0\)

Norm Axiom $\text N 1$: Positive Definiteness

$\Box$

Norm Axiom $\text N 2$: Multiplicativity

\(\ds \forall x, y \in S: \, \)

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

\(=\)

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

Definition of $\norm {\, \cdot \,}_S$

\(\ds \)

\(=\)

\(\ds \norm x \norm y\)

Norm Axiom $\text N 2$: Multiplicativity

\(\ds \)

\(=\)

\(\ds \norm x_S \norm y_S\)

Definition of $\norm {\, \cdot \,}_S$

$\Box$

Norm Axiom $\text N 3$: Triangle Inequality

\(\ds \forall x, y \in S: \, \)

\(\ds \norm {x + y}_S\)

\(=\)

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

Definition of $\norm {\, \cdot \,}_S$

\(\ds \)

\(\le\)

\(\ds \norm x + \norm y\)

Norm Axiom $\text N 3$: Triangle Inequality

\(\ds \)

\(=\)

\(\ds \norm x_S + \norm y_S\)

Definition of $\norm {\, \cdot \,}_S$

$\blacksquare$