View source for Functional (mathematics)

{{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]].

==Details==

===Duality===
The mapping
<math display=block>x_0 \mapsto f(x_0)</math>
is a function, where <math>x_0</math> is an [[argument of a function]] <math>f.</math> 
At the same time, the mapping of a function to the value of the function at a point
<math display=block>f \mapsto f(x_0)</math>
is a ''functional''; here, <math>x_0</math> is a [[parameter]].

Provided that <math>f</math> is a linear function from a vector space to the underlying scalar field, the above linear maps are [[Duality (mathematics)|dual]] to each other, and in functional analysis both are called [[linear functional]]s.

===Definite integral===
[[Integral]]s such as
<math display="block">f\mapsto I[f] = \int_{\Omega} H(f(x),f'(x),\ldots) \; \mu(\mathrm{d}x)</math>
form a special class of functionals. They map a function <math>f</math> into a real number, provided that <math>H</math> is real-valued. Examples include
* the area underneath the graph of a positive function <math>f</math> <math display=block>f\mapsto\int_{x_0}^{x_1}f(x)\;\mathrm{d}x</math>
* [[Lp norm|<math>L^p</math> norm]] of a function on a set <math>E</math> <math display=block>f\mapsto \left(\int_E|f|^p \; \mathrm{d}x\right)^{1/p}</math>
* the [[arclength]] of a curve in 2-dimensional Euclidean space <math display=block>f \mapsto \int_{x_0}^{x_1} \sqrt{ 1+|f'(x)|^2 } \; \mathrm{d}x</math>

===Inner product spaces===
Given an [[inner product space]] <math>X,</math> and a fixed vector <math>\vec{x} \in X,</math> the map defined by <math>\vec{y} \mapsto \vec{x} \cdot \vec{y}</math> is a linear functional on <math>X.</math> The set of vectors <math>\vec{y}</math> such that <math>\vec{x}\cdot \vec{y}</math> is zero is a vector subspace of <math>X,</math> called the ''null space'' or ''[[Kernel (linear algebra)|kernel]]'' of the functional, or the [[orthogonal complement]] of <math>\vec{x},</math> denoted <math>\{\vec{x}\}^\perp.</math>

For example, taking the inner product with a fixed function <math>g \in L^2([-\pi,\pi])</math> defines a (linear) functional on the [[Hilbert space]] <math>L^2([-\pi,\pi])</math> of square integrable functions on <math>[-\pi,\pi]:</math>
<math display=block>f \mapsto \langle f,g \rangle = \int_{[-\pi,\pi]} \bar{f} g</math>

===Locality===
If a functional's value can be computed for small segments of the input curve and then summed to find the total value, the functional is called local. Otherwise it is called non-local. For example:
<math display=block>F(y) = \int_{x_0}^{x_1}y(x)\;\mathrm{d}x</math>
is local while
<math display=block>F(y) = \frac{\int_{x_0}^{x_1}y(x)\;\mathrm{d}x}{\int_{x_0}^{x_1} (1+ [y(x)]^2)\;\mathrm{d}x}</math>
is non-local. This occurs commonly when integrals occur separately in the numerator and denominator of an equation such as in calculations of center of mass.

==Functional equations==
{{Main|Functional equation}}

The traditional usage also applies when one talks about a functional equation, meaning an equation between functionals: an equation <math>F = G</math> between functionals can be read as an 'equation to solve', with solutions being themselves functions. In such equations there may be several sets of variable unknowns, like when it is said that an [[Additive map|''additive'' map]] <math>f</math> is one ''satisfying [[Cauchy's functional equation]]'': 
<math display=block>f(x + y) = f(x) + f(y) \qquad \text{ for all } x, y.</math>

==Derivative and integration==
{{see also|Calculus of variations}}
[[Functional derivative]]s are used in [[Lagrangian mechanics]]. They are derivatives of functionals; that is, they carry information on how a functional changes when the input function changes by a small amount.

[[Richard Feynman]] used [[Functional integration|functional integrals]] as the central idea in his [[Path integral formulation|sum over the histories]] formulation of [[quantum mechanics]]. This usage implies an integral taken over some [[function space]].

== See also ==

* {{annotated link|Linear form}}
* {{annotated link|Optimization (mathematics)}}
* {{annotated link|Tensor}}

==References==

{{reflist}}
{{reflist|group=note}}

* {{Citation|last=Axler|first=Sheldon|title=Linear Algebra Done Right|volume=|pages=|publication-date=2015|series=[[Undergraduate Texts in Mathematics]]|date=December 18, 2014 |edition=3rd|publisher=[[Springer Science+Business Media|Springer]]|isbn=978-3-319-11079-0|author-link=Sheldon Axler}}
* {{Kolmogorov Fomin Elements of the Theory of Functions and Functional Analysis}} <!--{{sfn|Kolmogorov|Fomin|1957|p=}}-->
* {{Lang Algebra|edition=3r|pages=142&ndash;146 |chapter=III. Modules, §6. The dual space and dual module}}
* {{Wilansky Topology for Analysis 2008}} <!--{{sfn|Wilansky|2008|p=}}-->
* {{springer|title=Functional|id=p/f042010|author-last=Sobolev|author-first=V.I.}}
* {{nlab|id=linear+functional|title=Linear functional}} 
* {{nlab|id=nonlinear+functional|title=Nonlinear functional}} 
* {{MathWorld|title=Functional|urlname=Functional|author=Rowland, Todd}}
* {{MathWorld|title=Linear functional|urlname=LinearFunctional|author=Rowland, Todd}}

{{Authority control}}

[[Category:Types of functions]]