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 field of scalars. 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 given functional. A particularly important application in physics is searching for a state of a system that minimizes the energy functional.

The mapping

is a function, where x₀ is an argument of a function f. At the same time, the mapping of a function to the value of the function at a point

is a functional; here, x₀ is a parameter.

Provided that f is a linear function from a vector space to the underlying scalar field, the above linear maps are dual to each other, and in functional analysis both are called linear functionals.

Integrals such as

form a special class of functionals. They map a function ${\displaystyle f}$ into a real number, provided that ${\displaystyle H}$ is real-valued. Examples include

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