# Definition:Supremum of Mapping/Real-Valued Function

## Definition

Let $f: S \to \R$ be a real-valued function.

Let $f$ be bounded above on $S$.

### Definition 1

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

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

where

$\sup f \sqbrk S$ is the supremum in $\R$ of the image of $S$ under $f$.

### Definition 2

The supremum of $f$ on $S$ is defined as $\ds \sup_{x \mathop \in S} \map f x := K \in \R$ such that:

$(1): \quad \forall x \in S: \map f x \le K$
$(2): \quad \exists x \in S: \forall \epsilon \in \R_{>0}: \map f x > K - \epsilon$

## 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.