# Definition:Bounded Mapping/Normed Division Ring

*This page is about mappings to normed division rings which are bounded. For other uses, see Definition:Bounded.*

## Definition

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** if and only if 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 $s \in S$.

## Also see

- Metric Induced by Norm is Metric: this definition coincides with the definition of a bounded mapping to a metric space, using the norm on $R$.