# Definition:Supremum of Mapping

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

## Contents

## 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:

- $\displaystyle \sup_{x \mathop \in S} \map f x := \sup f \sqbrk S$

where

## 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.