Supremum

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In mathematics, given a subset S of a partially ordered set T, the supremum (sup) of S, if it exists, is the least element of T that is greater than or equal to each element of S. Consequently, the supremum is also referred to as the least upper bound (lub or LUB). If the supremum exists, it may or may not belong to S. If the supremum exists, it is unique.

Suprema are often considered for subsets of real numbers, rational numbers, or any other well-known mathematical structure for which it is immediately clear what it means for an element to be "greater-than-or-equal-to" another element. The definition generalizes easily to the more abstract setting of order theory, where one considers arbitrary partially ordered sets.

The concept of supremum is not the same as the concepts of minimal upper bound, maximal element, or greatest element.

Contents

Supremum of a set of real numbers

In analysis, the supremum or least upper bound of a set S of real numbers is denoted by sup(S) and is defined to be the smallest real number that is greater than or equal to every number in S. An important property of the real numbers is its completeness: every nonempty subset of the set of real numbers that is bounded above has a supremum that is also a real number.

Examples

In the last example, the supremum of a set of rationals is irrational, which means that the rationals are incomplete.

One basic property of the supremum is

for any functionals f and g.

If, in addition, we define sup(S) = −∞ when S is empty and sup(S) = +∞ when S is not bounded above, then every set of real numbers has a supremum under the affinely extended real number system.

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