Definition:Infimum of Set/Real Numbers

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This page is about infima of subsets of the real numbers. For other uses, see Definition:Infimum.

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:

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

admits an infimum:

$\inf S = 1$


Example 2

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

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

admits an infimum:

$\inf T = 1$


Example 3

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

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

admits an infimum:

$\inf V = 0$


Also see


Linguistic Note

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


Sources