# Definition:Infimum of Mapping/Real-Valued Function

## Definition

Let $f: S \to \R$ be a real-valued function.

Let $f$ be bounded below on $S$.

### Definition 1

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 $f$.

### Definition 2

The infimum of $f$ on $S$ is defined as $\ds \inf_{x \mathop \in S} \map f x := k \in \R$ such that:

$(1): \quad \forall x \in S: k \le \map f x$
$(2): \quad \forall \epsilon \in \R_{>0}: \exists x \in S: \map f x < k + \epsilon$

## 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.