Functional (mathematics)

In mathematics, a functional 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 a linear form, which is a linear mapping from a vector space V {\displaystyle V} into its field of scalars (that is, it is an element of the dual space V ∗ {\displaystyle V^{*}} ) In functional analysis and related fields, it refers to a mapping from a space X {\displaystyle X} into the field of real or complex numbers. In functional analysis, the term linear functional is a synonym of linear form; 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 . {\displaystyle X.}

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. 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 {\displaystyle X} is a space of functions, 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 X {\displaystyle 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 ↦ f ( x 0 ) {\displaystyle x_{0}\mapsto f(x_{0})}

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

f ↦ f ( x 0 ) {\displaystyle f\mapsto f(x_{0})}

is a functional; here, x 0 {\displaystyle x_{0}} is a parameter. Provided that f {\displaystyle 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 ↦ I [ f ] = ∫ Ω H ( f ( x ) , f ′ ( x ) , … ) μ ( d x ) {\displaystyle 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 {\displaystyle f} into a real number, provided that H {\displaystyle H} is real-valued. Examples include

the area underneath the graph of a positive function f {\displaystyle f} f ↦ ∫ x 0 x 1 f ( x ) d x {\displaystyle f\mapsto \int _{x_{0}}^{x_{1}}f(x)\;\mathrm {d} x}

L p {\displaystyle L^{p}} norm of a function on a set E {\displaystyle E} f ↦ ( ∫ E | f | p d x ) 1 / p {\displaystyle f\mapsto \left(\int _{E}|f|^{p}\;\mathrm {d} x\right)^{1/p}}

… excerpt ends here. Continue reading the full article.