Maximum

In mathematics, the maximum of a set S is the greatest element among its members. If there exists an element m in S such that m ≥ x for every x in S, then m is a maximum of S. If such an element does not exist, S has no maximum. The maximum, when it exists, is unique.

The concepts of maximum and supremum are related in the real numbers and other ordered fields. The supremum (least upper bound) of a set may exist even if the set has no maximum. If the supremum lies within the set, it is the maximum; otherwise, the set has a supremum but no maximum.

Examples illustrate the ideas. The set {2, -7, 3} has maximum 3. The set of natural numbers {1, 2, 3, …} has no maximum, since it is unbounded above. The open interval (0, 1) has no maximum because 1 is not included in the interval.

In function analysis and optimization, the maximum of a function f on a domain D is the value max_{x in D} f(x) if such a value exists. If there is a point x* in D with f(x*) ≥ f(x) for all x in D, then f achieves its maximum at x*, and x* is a maximizer. If no such x* exists but values approach a bound, the maximum does not exist, though a supremum may.

In practice, the term maximum also describes upper bounds or observed greatest values, such as the maximum score attained, the maximum rating of a device, or the value that maximizes a criterion in statistical or optimization contexts.