Definition:Infimum of Set/Real Numbers

From ProofWiki
Jump to navigation Jump to search

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

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


Some sources introduce the notation $\displaystyle \inf_{y \mathop \in S} y$, which may improve clarity in some circumstances.


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