Bounded linear operator

A bounded linear operator is a rule that takes inputs from one mathematical space and transforms them into outputs in another space, following two key principles: linearity (preserving addition and scalar multiplication) and boundedness (limiting how much outputs can grow compared to inputs). In simple terms, it acts like a controlled, predictable function that doesn’t produce wildly large results from reasonable inputs. These operators are fundamental in advanced mathematics and physics because they reliably relate different spaces, ensuring stability and continuity in calculations and models.