# Definition:Upper Bound of Mapping

*This page is about upper bounds of mappings which are bounded above. For other uses, see Definition:Upper Bound.*

## Definition

Let $f: S \to T$ be a mapping whose codomain is an ordered set $\left({T, \preceq}\right)$.

Let $f$ be bounded above in $T$ by $H \in T$.

Then $H$ is an **upper bound of $f$**.

### Real-Valued Function

The concept is usually encountered where $\left({T, \preceq}\right)$ is the set of real numbers under the usual ordering $\left({\R, \le}\right)$:

Let $f: S \to \R$ be a real-valued function.

Let $f$ be bounded above in $\R$ by $H \in \R$.

Then $H$ is an **upper bound of $f$**.