Functional (mathematics)
In mathematics, the term functional (as a noun) has at least three meanings.
- In modern linear algebra, it refers to a linear mapping from a vector space into its field of scalars, i.e., it refers to an element of the dual space .
- In mathematical analysis, more generally and historically, it refers to a mapping from a space into the real numbers, or sometimes into the complex numbers, for the purpose of establishing a calculus-like structure on . Depending on the author, such mappings may or may not be assumed to be linear, or to be defined on the whole space .
- In computer science, it is synonymous with higher-order functions, i.e. functions that take functions as argument or return 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 article on higher-order functions.
Commonly, the space is a space of functions. In this case, the functional is a "function of a function", 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 in the calculus of variations, where one searches for a function that minimizes a given functional. A particularly important application in physics is searching for a state of a system that minimizes the energy functional.
Contents
Details[edit]
Duality[edit]
The mapping
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
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[edit]
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
- L^{p} norm of a function on a set
- the arclength of a curve in 2-dimensional Euclidean space
Inner product spaces[edit]
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 :
Locality[edit]
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.
Equation solving[edit]
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 function f is one satisfying the functional equation
Derivative and integration[edit]
Functional derivatives are used in Lagrangian mechanics. They are derivatives of functionals: i.e. 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 also[edit]
References[edit]
- Hazewinkel, Michiel, ed. (2001) [1994], "Functional", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Rowland, Todd. "Functional". MathWorld.
- Lang, Serge (2002), "III. Modules, §6. The dual space and dual module", Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, pp. 142–146, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001