Definition:Infimum of Set/Real Numbers

Definition

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 $\R$.

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

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

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 $\map {\operatorname {glb} } T$ or $\map {\operatorname {g.l.b.} } T$.

Some sources refer to the infimum of a set as the infimum on a set.

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.

Examples

Example 1

The subset $S$ of the real numbers $\R$ defined as:

$S = \set {1, 2, 3}$

$\inf S = 1$

Example 2

The subset $T$ of the real numbers $\R$ defined as:

$T = \set {x \in \R: 1 \le x \le 2}$

$\inf T = 1$

Example 3

The subset $V$ of the real numbers $\R$ defined as:

$V := \set {x \in \R: x > 0}$

$\inf V = 0$