# Definition:Infimum of Set

## Definition

Let $\struct {S, \preceq}$ be an ordered set.

Let $T \subseteq S$.

An element $c \in S$ is the infimum of $T$ in $S$ if and only if:

$(1): \quad c$ is a lower bound of $T$ in $S$
$(2): \quad d \preceq c$ for all lower bounds $d$ of $T$ in $S$.

If there exists an infimum of $T$ (in $S$), we say that $T$ admits an infimum (in $S$).

### Subset of Real Numbers

The concept is usually encountered where $\struct {S, \preceq}$ is the set of real numbers under the usual ordering $\struct {\R, \le}$:

Let $T \subseteq \R$.

A real number $c \in \R$ is the infimum of $T$ in $\R$ if and only if:

$(1): \quad c$ is a lower bound of $T$ in $\R$
$(2): \quad d \le c$ for all lower bounds $d$ of $T$ in $\R$.

The infimum of $T$ is denoted $\inf T$ or $\map \inf T$.

### Finite Infimum

If $T$ is finite, $\inf T$ is called a finite infimum.

## 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 $\displaystyle \inf_{y \mathop \in S} y$, which may improve clarity in some circumstances.

Some older sources, applying the concept to a (strictly) decreasing real sequence, refer to a 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.