Definition:Infimum of Mapping
This page is about Infimum in the context of Mapping. For other uses, see Infimum.
Definition
Let $S$ be a set.
Let $\struct {T, \preceq}$ be an ordered set.
Let $f: S \to T$ be a mapping from $S$ to $T$.
Let $f \sqbrk S$, the image of $f$, admit an infimum.
Then the infimum of $f$ (on $S$) is defined by:
- $\ds \inf_{x \mathop \in S} \map f x = \inf f \sqbrk S$
Real-Valued Function
The infimum of $f$ on $S$ is defined by:
- $\ds \inf_{x \mathop \in S} \map f x = \inf f \sqbrk S$
where
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 $\map {\mathrm {glb} } T$ or $\map {\mathrm {g.l.b.} } T$.
Some sources refer to the infimum of a set as the infimum on a set.
Some sources introduce the notation $\ds \inf_{y \mathop \in S} y$, which may improve clarity in some circumstances.
Some older sources, applying the concept to a (strictly) decreasing real sequence, refer to an infimum as a lower limit.
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.