Editing Functional (mathematics) (section)

{{Short description|Types of mappings in mathematics}}
{{Other uses|Functional (disambiguation){{!}}Functional}}
{{Distinguish|functional notation}}
{{More citations needed|date=September 2023}}

{{Use American English|date = February 2019}}
{{Use mdy dates|date = February 2019}}

[[File:Arclength.svg|400px|right|thumb|The [[arc length]] functional has as its domain the vector space of [[rectifiable curve]]s &ndash; a subspace of <math>C([0,1],\R^3)</math> &ndash; and outputs a real scalar. This is an example of a non-linear functional.]]
[[File:Integral as region under curve.svg|thumb|right|The [[Riemann integral]] is a [[linear functional]] on the vector space of functions defined on {{math|[''a'', ''b'']}} that are  Riemann-integrable from {{mvar|a}} to {{mvar|b}}.]]

In [[mathematics]], a '''functional''' is a certain type of [[Function (mathematics)|function]]. The exact definition of the term varies depending on the subfield (and sometimes even the author).
* In [[linear algebra]], it is synonymous with a [[linear form]], which is a linear mapping from a vector space <math>V</math> into its [[Field (mathematics)|field of scalars]] (that is, it is an element of the [[dual space]] <math>V^*</math>)<ref name=LangAlgebra2002DefFunctional>{{harvnb|Lang|2002|p=142}} "Let ''E'' be a free module over a commutative ring ''A''. We view ''A'' as a free module of rank 1 over itself. By the '''dual module''' ''E''<sup>∨</sup> of ''E'' we shall mean the module Hom(''E'', ''A''). Its elements will be called '''functionals'''. Thus a functional on ''E'' is an ''A''-linear map ''f'' : ''E'' → ''A''."</ref>
* In [[functional analysis]] and related fields, it refers to a mapping from a space <math>X</math> into the field of [[Real numbers|real]] or [[complex numbers]].<ref name=KolmogorovDefFunctionalOnLinearSpace>{{harvnb|Kolmogorov|Fomin|1957|p=77}} "A numerical function ''f''(''x'') defined on a normed linear space ''R'' will be called a ''functional''. A functional ''f''(''x'') is said to be ''linear'' if ''f''(α''x'' + β''y'') = α''f''(''x'') + β''f''(''y'') where ''x'', ''y'' ∈ ''R'' and α, β are arbitrary numbers."</ref>{{sfn|Wilansky|2008|p=7}} In functional analysis, the term {{em|[[linear functional]]}} is a synonym of [[linear form]];{{sfn|Wilansky|2008|p=7}}<ref name=Axler2015>{{Harvard citation text|Axler|2014}} p. 101, §3.92</ref><ref name=EOFLinearFunctional>{{springer|title=Linear functional|oldid=51214|author-last=Khelemskii|author-first=A.Ya.}}</ref> that is, it is a scalar-valued linear map. Depending on the author, such mappings may or may not be assumed to be linear, or to be defined on the whole space <math>X.</math>{{citation needed|date=December 2021}}
* In [[computer science]], it is synonymous with a [[higher-order function]], which is a function that takes one or more functions as arguments or returns them.{{citation needed|date=August 2024}}

This article is mainly concerned with the second concept, which arose in the early 18th century as part of the [[calculus of variations]]. The first concept, which is more modern and abstract, is discussed in detail in a separate article, under the name [[linear form]]. The third concept is detailed in the computer science article on [[higher-order function]]s.

In the case where the space <math>X</math> is a space of functions, the functional is a "function of a function",<ref name=KolmogorovDefFunctionalAsMapDefinedOnSetOfFunctions>{{harvnb|Kolmogorov|Fomin|1957|loc=pp. 62-63 "A real function on a space ''R'' is a mapping of ''R'' into the space ''R''<sup>1</sup> (the real line). Thus, for example, a mapping of ''R''<sup>''n''</sup> into ''R''<sup>1</sup> is an ordinary real-valued function of ''n'' variables. In the case where the space ''R'' itself consists of functions, the functions of the elements of ''R'' are usually called ''functionals''."}}</ref> and some older authors actually define the term "functional" to mean "function of a function".
However, the fact that <math>X</math> is a space of functions is not mathematically essential, so this older definition is no lo<!-- So what is the new definition? They seem to be the same -->nger prevalent.{{Citation needed|date=January 2019}}

The term originates from the [[calculus of variations]], where one searches for a function that minimizes (or maximizes) a given functional. A particularly important application in [[physics]] is search for a state of a system that minimizes (or maximizes) the [[Action (physics)|action]], or in other words the time integral of the [[Lagrangian mechanics#Introduction|Lagrangian]].