Functional (mathematics)

In mathematics, and particularly in functional analysis and the calculus of variations, a functional is a function from a vector space into its underlying scalar field, or a set of functions of the real numbers. In other words, it is a function that takes a vector as its input argument, and returns a scalar. Commonly the vector space is a space of functions, thus the functional takes a function for its input argument, then it is sometimes considered a function of a function (a higher-order function). Its use originates in the calculus of variations where one searches for a function that minimizes a certain functional. A particularly important application in physics is searching for a state of a system that minimizes the energy functional.