Definition:Supremum

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Disambiguation

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Supremum may refer to:

• Supremum of Ordered Set
• Supremum of Subset of Real Numbers

• Supremum of Mapping
• Supremum of Real-Valued Function

• Supremum of Sequence
• Supremum of Real Sequence

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.

Also see

• Definition:Bounded Above
• Definition:Upper Bound

• Definition:Bounded Below
• Definition:Lower Bound

• Definition:Bounded
• Definition:Bound

• Definition:Infimum

• Definition:Unbounded
• Definition:Unbounded Above
• Definition:Unbounded Below

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): least upper bound (l.u.b.; supremum)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): least upper bound (l.u.b.; supremum)