# 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.