Functional (mathematics)

In mathematics, a functional (as a noun) 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 linear forms, which are linear mapping from a vector space $V$ into its field of scalars (that is, an element of the dual space $V^{*}$ )[1]
In functional analysis and related fields, it refers more generally 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 $X.$
In computer science, it is synonymous with higher-order functions, that is, functions that take functions as arguments or return them.

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.

The Riemann integral is a linear functional on the vector space of functions defined on

[a, b]

that are Riemann-integrable from

a

to

b

.

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.

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.

Details

Duality

The mapping

x_{0}\mapsto f(x_{0})

is a function, where $x_{0}$ is an argument of a function $f.$ At the same time, the mapping of a function to the value of the function at a point

f\mapsto f(x_{0})

is a functional; here, $x_{0}$ is a parameter.

Provided that $f$ is a linear function from a vector space to the underlying scalar field, the above linear maps are dual to each other, and in functional analysis both are called linear functionals.

Definite integral

Integrals such as

f\mapsto I[f]=\int _{\Omega }H(f(x),f'(x),\ldots )\;\mu (\mathrm {d} x)

form a special class of functionals. They map a function $f$ into a real number, provided that $H$ is real-valued. Examples include

the area underneath the graph of a positive function $f$ $f\mapsto \int _{x_{0}}^{x_{1}}f(x)\;\mathrm {d} x$
$L^{p}$ norm of a function on a set $E$ $f\mapsto \left(\int _{E}|f|^{p}\;\mathrm {d} x\right)^{1/p}$
the arclength of a curve in 2-dimensional Euclidean space $f\mapsto \int _{x_{0}}^{x_{1}}{\sqrt {1+|f'(x)|^{2}}}\;\mathrm {d} x$

Inner product spaces

Given an inner product space $X,$ and a fixed vector ${\vec {x}}\in X,$ the map defined by ${\vec {y}}\mapsto {\vec {x}}\cdot {\vec {y}}$ is a linear functional on $X.$ The set of vectors ${\vec {y}}$ such that ${\vec {x}}\cdot {\vec {y}}$ is zero is a vector subspace of $X,$ called the null space or kernel of the functional, or the orthogonal complement of ${\vec {x}},$ denoted $\{{\vec {x}}\}^{\perp }.$

For example, taking the inner product with a fixed function $g\in L^{2}([-\pi ,\pi ])$ defines a (linear) functional on the Hilbert space $L^{2}([-\pi ,\pi ])$ of square integrable functions on $[-\pi ,\pi ]:$

f\mapsto \langle f,g\rangle =\int _{[-\pi ,\pi ]}{\bar {f}}g

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:

F(y)=\int _{x_{0}}^{x_{1}}y(x)\;\mathrm {d} x

is local while

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}}

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

The traditional usage also applies when one talks about a functional equation, meaning an equation between functionals: an equation $F=G$ 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 $f$ is one satisfying Cauchy's functional equation:

f(x+y)=f(x)+f(y)\qquad {\text{ for all }}x,y.

Derivative and integration

Functional derivatives 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 integrals as the central idea in his sum over the histories formulation of quantum mechanics. This usage implies an integral taken over some function space.

References

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."
Wilansky 2013, p. 7.
Axler (2015) 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."

Axler, Sheldon (December 18, 2014), Linear Algebra Done Right, Undergraduate Texts in Mathematics (3rd ed.), Springer (published 2015), ISBN 978-3-319-11079-0
Kolmogorov, Andrey; Fomin, Sergei V. (1957). Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics. New York: Dover Books. ISBN 978-1-61427-304-2. OCLC 912495626.
Lang, Serge (2002), "III. Modules, §6. The dual space and dual module", Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, pp. 142–146, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001
Wilansky, Albert (October 17, 2008) [1970]. Topology for Analysis. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-46903-4. OCLC 227923899.
Sobolev, V.I. (2001) [1994], "Functional", Encyclopedia of Mathematics, EMS Press
Linear functional at the nLab
Nonlinear functional at the nLab
Rowland, Todd. "Functional". MathWorld.
Rowland, Todd. "Linear functional". MathWorld.

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[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."

[FOOTNOTEWilansky20137-3] Wilansky 2013, p. 7.

[Axler2015-4] Axler (2015) 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."