Definition:Bounded Above Mapping

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

Then $f$ is bounded above on $S$ by the upper bound $H$ iff:
 * $\forall x \in S: f \left({x}\right) \preceq H$

That is, iff $f \left({S}\right) = \left\{{f \left({x}\right): x \in S}\right\}$ is bounded above by $H$.

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

Also see

 * Definition:Upper Bound of Mapping


 * Definition:Bounded Below Mapping
 * Definition:Lower Bound of Mapping


 * Definition:Bounded Mapping