Functional (mathematics)

The arc length functional has as its domain the vector space of rectifiable curves – a subspace of $C([0,1],\mathbb {R} ^{3})$ – and outputs a real scalar. This is an example of a non-linear functional.

In mathematics, a functional is a certain type of 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 $V$ into its field of scalars (that is, it is an element of the dual space $V^{*}$ )^[1]
In functional analysis and related fields, it refers to a mapping from a space $X$ into the field of real or complex numbers.^[2]^[3] In functional analysis, the term linear functional is a synonym of linear form;^[3]^[4]^[5] 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 $X.$ ^{[citation needed]}
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.

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 functions.

In the case where the space $X$ is a space of functions, the functional is a "function of a function",^[6] and some older authors actually define the term "functional" to mean "function of a function". However, the fact that $X$ is a space of functions is not mathematically essential, so this older definition is no longer prevalent.^{[citation needed]}

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, or in other words the time integral of the Lagrangian.

^ 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^∨ 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."
^ 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(αx + βy) = αf(x) + βf(y) where x, y ∈ R and α, β are arbitrary numbers."
^ ^a ^b Wilansky 2008, p. 7.
^ Axler (2014) p. 101, §3.92
^ Khelemskii, A.Ya. (2001) [1994], "Linear functional", Encyclopedia of Mathematics, EMS Press
^ Kolmogorov & Fomin 1957, pp. 62-63 "A real function on a space R is a mapping of R into the space R¹ (the real line). Thus, for example, a mapping of Rⁿ into R¹ 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."

[LangAlgebra2002DefFunctional-1] 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^∨ 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."

[KolmogorovDefFunctionalOnLinearSpace-2] 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(αx + βy) = αf(x) + βf(y) where x, y ∈ R and α, β are arbitrary numbers."

[FOOTNOTEWilansky20087-3] Wilansky 2008, p. 7.

[Axler2015-4] Axler (2014) p. 101, §3.92

[EOFLinearFunctional-5] Khelemskii, A.Ya. (2001) [1994], "Linear functional", Encyclopedia of Mathematics, EMS Press

[KolmogorovDefFunctionalAsMapDefinedOnSetOfFunctions-6] Kolmogorov & Fomin 1957, pp. 62-63 "A real function on a space R is a mapping of R into the space R¹ (the real line). Thus, for example, a mapping of Rⁿ into R¹ 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."

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