Definition:Infimum of Mapping

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This page is about infima of mappings. For other uses, see Definition:Infimum.

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 infimum.


Then the infimum of $f$ (on $S$) is defined by:

$\displaystyle \inf_{x \mathop \in S} f \left({x}\right) = \inf f \left[{S}\right]$


Real-Valued Function

The infimum of $f$ on $S$ is defined by:

$\displaystyle \inf_{x \mathop \in S} f \left({x}\right) = \inf f \left[{S}\right]$

where

$\inf f \left[{S}\right]$ is the infimum in $\R$ of the image of $S$ under $f$.


Also known as

Particularly in the field of analysis, the infimum of a set $T$ is often referred to as the greatest lower bound of $T$ and denoted $\map {\operatorname {glb} } T$ or $\map {\operatorname {g.l.b.} } T$.

Some sources refer to the infimum of a set as the infimum on a set.


Also defined as

Some sources refer to the infimum as being the lower bound.

Using this convention, any element less than this is not considered to be a lower bound.


Linguistic Note

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


Also see