Definition:Normed Division Subring

Definition

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

A normed division subring of $\struct {R, +, \circ, \norm {\, \cdot \,} }$ is a subset $S$ of $R$ such that $\struct{S, +_S, \circ_S, \norm{\, \cdot \,}_S}$ is a normed division ring where:

$(1) \quad +_S$ is the binary operation $+$ restricted to $S \times S$

$(2) \quad \circ_S$ is the binary operation $\circ$ restricted to $S \times S$

$(3) \quad \norm {\, \cdot \,}_S$ is the norm $\norm {\, \cdot \,}$ restricted to $S$.