Definition:Bounded Mapping/Normed Division Ring

From ProofWiki
Jump to navigation Jump to search

This page is about Bounded Mapping in the context of Normed Division Ring. For other uses, see 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