Definition:Supremum of Mapping

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This page is about suprema of mappings. For other uses, see Definition:Supremum.

Definition

Let $S$ be a set.

Let $ \left({T, \preceq}\right)$ be an ordered set.

Let $f: S \to T$ be a mapping from $S$ to $T$.

Let $f \left[{S}\right]$, the image of $f$, admit a supremum.


Then the supremum of $f$ (on $S$) is defined by:

$\displaystyle \sup_{x \mathop \in S} f \left({x}\right) = \sup f \left[{S}\right]$


Real-Valued Function

The supremum of $f$ on $S$ is defined by:

$\displaystyle \sup_{x \mathop \in S} f \left({x}\right) := \sup f \left[{S}\right]$

where

$\sup f \left[{S}\right]$ is the supremum in $\R$ of the image of $S$ under $f$.


Also known as

Particularly in the field of analysis, the supremum of a set $T$ is often referred to as the least upper bound of $T$ and denoted $\map {\operatorname {lub} } T$ or $\map {\operatorname {l.u.b.} } T$.

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


Also defined as

Some sources refer to the supremum as being the upper bound.

Using this convention, any element greater than this is not considered to be an upper bound.


Linguistic Note

The plural of supremum is suprema, although the (incorrect) form supremums can occasionally be found if you look hard enough.


Also see