maksverdier

Maksverdier (plural of maksverdi) is a term used in Norwegian mathematics to denote the largest value that a quantity can attain within a given domain. In English, the corresponding concept is maximum values or maxima. A maksverdi exists for a set S if there is an element m in S such that m is greater than or equal to every element of S; in that case m is called the maximum of S. When no such element exists, one may still have upper bounds, and the least upper bound is called the supremum (or least upper bound); the supremum may lie outside the set S.

Formally, let S be a subset of a partially ordered set with ≤. The maximum of S is an element m in S with x ≤ m for all x in S. The supremum of S is the least element u such that x ≤ u for all x in S; if the supremum belongs to S, it is also the maximum. A set may have a maximum or, in some cases, no maximum at all, even though an upper bound exists.

Examples help illustrate: the finite set {2, 5, 7, 3} has maximum 7. The closed interval [0, 1] has maximum 1, while the open interval (0, 1) has no maximum but has supremum 1.

Applications of maksverdier occur in optimization, data analysis, economics, and decision theory, wherever it is necessary to identify the largest feasible outcome within constraints. See also: maximum, supremum, bound, attainment, extremum.