# Definition:Supremum of Mapping

## 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.