bounded linear operators

A bounded linear operator is a mathematical function that takes elements from one space and transforms them into another, respecting two key properties: linearity and boundedness. Linearity means the operator preserves addition and scalar multiplication, so applying the operator to combined inputs is the same as combining their individual outputs. Boundedness means there's a limit to how much the output can grow relative to the input size, ensuring the transformation doesn't produce infinitely large results from finite inputs. Together, these properties make such operators predictable and manageable within mathematical analysis, especially in infinite-dimensional spaces.