- In linear algebra, it is synonymous with linear forms, which are linear mapping from a vector space into its field of scalars (that is, an element 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 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 
- 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.
Integrals such as
- 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 spaces
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
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:
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 integration
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.
- Linear form – Linear map from a vector space to its field of scalars
- Optimization (mathematics)
- Tensor – Algebraic object with geometric applications
- Lang 2002, p. 142 harvnb error: no target: CITEREFLang2002 (help) "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 : E → A."
- 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 f(αx + βy) = αf(x) βf(y) where x, y ∈ R and α, β are arbitrary numbers."
- Wilansky 2013, p. 7. sfn error: no target: CITEREFWilansky2013 (help)
- Axler (2015) harvtxt error: no target: CITEREFAxler2015 (help) p. 101, §3.92
- Khelemskii, A.Ya. (2001) , "Linear functional", Encyclopedia of Mathematics, EMS Press
- 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."
- Axler, Sheldon (December 18, 2014), Linear Algebra Done Right, Undergraduate Texts in Mathematics (3rd ed.), Springer (published 2015), ISBN 978-3-319-11079-0
- Kolmogorov, Andrey; Fomin, Sergei V. (1957). Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics. New York: Dover Books. ISBN 978-1-61427-304-2. OCLC 912495626.
- Lang, Serge (2002), "III. Modules, §6. The dual space and dual module", Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, pp. 142–146, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001
- Wilansky, Albert (October 17, 2008) . Topology for Analysis. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-46903-4. OCLC 227923899.
- Sobolev, V.I. (2001) , "Functional", Encyclopedia of Mathematics, EMS Press
- Linear functional at the nLab
- Nonlinear functional at the nLab
- Rowland, Todd. "Functional". MathWorld.
- Rowland, Todd. "Linear functional". MathWorld.