Definition:Infimum of Set

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This page is about Infimum in the context of Ordered Set. For other uses, see Infimum.

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 $\map {\operatorname {glb} } T$ 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.


Also see


Special cases


Linguistic Note

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


Sources