bounded operators

Bounded operators are functions that map elements from one space to another in a way that keeps outputs "controlled" within certain limits. In mathematical terms, they occur in the study of linear transformations between spaces (like spaces of numbers or functions) and ensure that when you apply the operator, the results don’t grow too large, maintaining a consistent relationship between inputs and outputs. This property is essential in various fields, including physics and engineering, as it guarantees stability and predictability in systems modeled by these operators.