/ ?2 f$ s" W5 ~In mathematics, a functional is traditionally a map from a vector space to the field underlying the vector space, which is usually the real numbers. In other words, it is a function that takes a vector as its argument or input and returns a scalar. Its use goes back to the calculus of variations where one searches for a function which minimizes a certain functional. A particularly important application in physics is to search for a state of a system which minimizes the energy functional.% F# C& |6 }4 U' H6 C1 g
In functional analysis, the functional is also used in a broader sense as a mapping from an arbitrary vector space into the underlying scalar field (usually, real or complex numbers). A special kind of such functionals, linear functionals, gives rise to the study of dual spaces. * j0 X; j- k' a5 e) b: mTransformations of functions is a somewhat more general concept, see operator.