Functional (mathematics)

Summary

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).

The arc length functional has as its domain the vector space of rectifiable curves – a subspace of – 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 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 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.

DetailsEdit

DualityEdit

The mapping

 
is a function, where   is an argument of a function   At the same time, the mapping of a function to the value of the function at a point
 
is a functional; here,   is a parameter.

Provided that   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 integralEdit

Integrals such as

 
form a special class of functionals. They map a function   into a real number, provided that   is real-valued. Examples include
  • the area underneath the graph of a positive function  
     
  •   norm of a function on a set  
     
  • the arclength of a curve in 2-dimensional Euclidean space
     

Inner product spacesEdit

Given an inner product space   and a fixed vector   the map defined by   is a linear functional on   The set of vectors   such that   is zero is a vector subspace of   called the null space or kernel of the functional, or the orthogonal complement of   denoted  

For example, taking the inner product with a fixed function   defines a (linear) functional on the Hilbert space   of square integrable functions on  

 

LocalityEdit

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:

 
is local while
 
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 equationsEdit

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

 

Derivative and integrationEdit

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.

See alsoEdit

ReferencesEdit

  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 : EA."
  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 fx + βy) = αf(x) βf(y) where x, yR and α, β are arbitrary numbers."
  3. ^ a b Wilansky 2013, p. 7.
  4. ^ Axler (2015) p. 101, §3.92
  5. ^ Khelemskii, A.Ya. (2001) [1994], "Linear functional", Encyclopedia of Mathematics, EMS Press
  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."