Functional (mathematics)

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In mathematics, the term functional is applied to certain functions. There are two common ways it is applied: these are related historically, but diverged somewhat during the twentieth century.

1 Original meaning
- 1.1 Functional equation
- 1.2 Functional derivative and functional integration
2 Linear functionals
3 See also
4 References

[edit] Original meaning

The initial meaning is a function that takes functions as its argument; that is, a function whose domain is a set of functions. This was how the word was used initially, in the calculus of variations, where the integral to be minimized should be a functional, applied to an as-yet unknown function satisfying only some boundary conditions, and differentiability conditions. (See also operator, for a somewhat broader concept.)

This usage still applies in that context and in many parts of physics and computer science, where in lambda calculus and functional programming a higher-order function is one that accepts a function and returns some value (or function).

An example of a functional, in the physicist's sense, is

$F[f]=\int_{-\infty}^{\infty}\mbox{d}Ef(E)$

which turns any integrable function $f$ into a real number, called the integral of $f$ .

[edit] Functional equation

The same 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, as when it is said that an additive function f is one satisfying the functional equation

f(x+y) = f(x) + f(y).

[edit] Functional derivative and functional integration

Functional derivatives are used in Lagrangian mechanics. They are derivatives of functionals: i.e. they carry information on how a functional changes, when the function changes by a small amount. See also calculus of variations.

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.

[edit] Linear functionals

The secondary usage in the compound linear functional arises from functional analysis. While in the foundational period of functional analysis from 1900-1920, it was largely the study of vector spaces such as the Lp spaces that are function spaces, the later axiomatic approach made no such assumption. The name linear functional, however, was carried over and applied to the dual space construction, in the general case.