Definition:Infimum of Set/Real Numbers

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This page is about Infimum of Subset of Real Numbers. For other uses, see 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 $\map {\mathrm {glb} } T$ or $\map {\mathrm {g.l.b.} } T$.

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

Some sources refer to the infimum of a set as the meet of the set and use the notation $\bigwedge T$ or $\ds \bigwedge_{y \mathop \in T} y$.


Some sources introduce the notation $\ds \inf_{y \mathop \in T} y$, 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.


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

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