Functional

A functional is a real- or complex-valued function on a vector space , usually of functions. For example, the energy functional on the unit disk assigns a finite number to a differentiable function whose gradient is square integrable,

For the functional to be continuous, it is necessary for the vector space of functions to have an appropriate topology. The widespread use of functionals in applications, such as the calculus of variations, gave rise to functional analysis.

The reason the term "functional" is used is because can be a space of functions, e.g.,

in which case is a linear functional on .