\begin{definition}[Definition:Bounded Mapping/Normed Division Ring]
Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring.
Let $f: S \to R$ be a mapping from $S$ into $R$.
Then $f$ is '''bounded''' {{iff}} the real-valued function $\norm {\,\cdot\,} \circ f: S \to \R$ is bounded, where $\norm {\,\cdot\,} \circ f$ is the composite of $\norm {\,\cdot\,}$ and $f$.
That is, $f$ is '''bounded''' if there is a constant $K \in \R_{\ge 0}$ such that $\norm{f \paren {s}} \le K$ for all $s \in S$.
\end{definition}