Linear Functional

A linear functional is a special kind of rule that takes a mathematical object called a vector and assigns it a single number, called a scalar. This rule is linear, meaning that if you add two vectors together or multiply a vector by a number, the scalar output behaves predictably: the sum of the results equals the result for the sum of the vectors, and multiplying a vector by a number scales its scalar accordingly. Linear functionals are useful in understanding and analyzing systems by capturing how they respond to different inputs in a consistent, linear manner.