Supremum

The supremum of a set in an ordered system is its least upper bound: an element at least as large as every member and no larger than necessary.

Partially ordered set: A set equipped with a partial order, a relation that is reflexive, antisymmetric, and transitive. A supremum is defined relative to an order, which need not compare every pair of elements.
Least upper bound property: The property that every nonempty subset of a specified kind, when bounded above, has a least upper bound. It states the completeness condition that ensures relevant suprema exist.
Supremum norm: A function norm defined by the supremum of the function’s absolute value over its domain. It measures the largest magnitude of a function, including when no point attains it.
Essential supremum: The smallest bound that holds almost everywhere with respect to a measure. It ignores exceptional measure-zero values, unlike an ordinary pointwise supremum.
Upper bound: An element greater than or equal to every member of a specified set in an ordered system. A supremum must first be an upper bound of the set.
Order relation: A binary relation that organizes elements by comparisons such as less than or equal to. The inequalities defining a supremum depend on the underlying order relation.
Uniform convergence: Convergence of functions in which one index threshold controls the approximation error across the entire domain. Its error bound is expressed as a supremum over all domain points.
Maximum: The greatest value attained by a function or contained in a set. A supremum is an upper bound even when no element attains it.
Least element: An element of an ordered set that is less than or equal to every element in that set. The supremum is least among the set’s upper bounds.
Rational numbers: The numbers expressible as a ratio of two integers with a nonzero denominator. Their missing least upper bounds expose the difference between ordered and complete systems.

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