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Concept
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).
In linear algebra, it is synonymous with linear forms, which are linear mappings from a vector space into its field of scalars (that is, they are elements of the dual space )
In functional analysis and related fields, it refers more generally to a mapping from a space into the field of real or complex numbers. In functional analysis, the term 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
In computer science, it is synonymous with higher-order functions, that is, functions that take functions as arguments 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 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", 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.
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.
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.
Official source
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