By Jacob Reis | Updated Aug 30, 2022
ClaudeLux/iStock/Getty Images
In mathematics, the terms bounded and unbounded appear across various subfields. Understanding their precise meanings helps avoid confusion, especially when they are applied to functions, operators, and sets.
A bounded function is one whose range lies between two finite limits. In a graph, this means the function’s values can be trapped by two horizontal lines. For instance, the sine function oscillates between –1 and 1, so it is bounded. Mathematically, a function f defined on a set X (with real or complex values) is bounded if there exists M > 0 such that |f(x)| ≤ M for every x ∈ X.
Conversely, an unbounded function has no such finite upper or lower limits; its values can grow arbitrarily large (or small). Functions like f(x) = 1/x (defined for x ≠ 0) or f(x) = x² are unbounded on their respective domains.
In functional analysis, operators act on elements of a vector space. An operator A is called bounded if there exists a constant C such that ‖A(x)‖ ≤ C‖x‖ for all x in its domain. If no such constant exists, the operator is unbounded. According to the Encyclopaedia of Mathematics, an unbounded operator maps a bounded set in its domain to an unbounded set in its codomain.
A set of numbers is bounded when it has both an upper and a lower finite bound. Classic examples include the interval [2, 401) and the sequence {1, ½, ⅓, ¼, …}. An unbounded set lacks at least one of these finite limits; for example, the set of all positive integers ℕ is unbounded because it has no finite upper bound.
