Definition:Infimum of Mapping

Definition
Let $S$ be a set.

Let $ \left({T, \preceq}\right)$ be an ordered set.

Let $f: S \to T$ be a mapping from $S$ to $T$.

Let $f \left[{S}\right]$, the image of $f$, admit an supremum.

Then the infimum of $f$ (on $S$) is defined by:
 * $\displaystyle \inf_{x \mathop \in S} f \left({x}\right) = \inf f \left[{S}\right]$

Also defined as
Some sources refer to the infimum as being the lower bound. Using this convention, any element greater than this is not considered to be a lower bound.

Also see

 * Definition:Supremum of Mapping

Linguistic Note
The plural of infimum is infima, although the (incorrect) form infimums can occasionally be found if you look hard enough.