Maxwert

Maxwert is the German term for the greatest value in mathematics and related disciplines. It refers to the largest element of a set or to the largest value that a function attains on its domain.

For a set S, a maximum (Maxwert) is an element m in S such that m ≥ s for every s in S. If such an element exists, S has a maximum and one writes max S = m. For a function f: D → R, the maximum value on D is max_{x in D} f(x), provided there exists x* in D with f(x*) = max_{x in D} f(x). The points x* where the maximum is achieved are called the argmax of f on D.

Supremum and maximum are related but distinct. The supremum (least upper bound) of a set S, denoted sup S, is the smallest number that is greater than or equal to every element of S. A maximum, when it exists, always equals the supremum; however, the supremum need not be attained within S. For example, on the open interval (0,1) the supremum is 1, but the maximum does not exist in the interval.

Existence results include the extreme value theorem: if a real-valued function is continuous on a compact (closed and bounded) domain, it attains a maximum. In finite sets or bounded domains, a maximum often exists directly.

Related concepts include the minimum value (Minwert), the minimum and maximum elements of sets, and the concept of argmax. The term is widely used in calculus, optimization, statistics, and computer science to denote the largest possible value under given conditions.