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

$\inf f \sqbrk S$ is the infimum in $\R$ of the image of $S$ under $f$.

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.

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 refer to the infimum of a set as the meet of the set and use the notation $\bigwedge T$ or $\ds \bigwedge_{y \mathop \in T} y$.

Some sources introduce the notation $\ds \inf_{y \mathop \in T} y$, 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.

Linguistic Note

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

Also see

• Definition:Supremum of Mapping

• Results about infima can be found here.