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 $S$