Definition:Bounded Above/Mapping

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Definition

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

Then $f$ is said to be bounded above (in $T \ $) 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 $H$.

If there is no such $H \in S$ then $f$ is unbounded above (in $T \ $).