# Definition:Supremum of Mapping

*This page is about Supremum in the context of Mapping. For other uses, see Supremum.*

## Definition

Let $S$ be a set.

Let $\struct {T, \preceq}$ be an ordered set.

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

Let $f \sqbrk S$, the image of $f$, admit a supremum.

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

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

### Real-Valued Function

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

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

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 $\map {\mathrm {lub} } T$ or $\map {\mathrm {l.u.b.} } T$.

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

Some sources refer to the **supremum of a set** as the **join of the set** and use the notation $\bigvee S$.

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

Some older sources, applying the concept to a (strictly) increasing real sequence, refer to a **supremum** as an **upper limit**.

## 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.