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In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar. Commonly the vector space is a space of functions, thus the functional takes a function for its input argument, then it is sometimes considered a function of a function. Its use originates in the calculus of variations where one searches for a function that minimizes a certain functional. A particularly important application in physics is searching for a state of a system that minimizes the energy functional.

Transformations of functions is a rather more general concept, see Operator (mathematics).

Examples
Duality

Observe that the mapping

$$x_0\mapsto f(x_0)$$

is a function, here $$x_0$$ 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

$$f\mapsto f(x_0)$$

is a functional, here $$x_0$$ is a parameter.

Provided that f is a linear function from a linear 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

$$f\mapsto I[f]=\int_{\Omega} H(f(x),f'(x),\ldots)\;\mu(\mbox{d}x)$$

form a special class of functionals. They map a function f into a real number, provided that H is real-valued. Examples include

the area underneath the graph of a positive function f

$$f\mapsto\int_{x_0}^{x_1}f(x)\;\mathrm{d}x$$

Lp norm of functions

$$f\mapsto \left(\int|f|^p \; \mathrm{d}x\right)^{1/p}$$

the arclength of a curve in 2-dimensional Euclidean space

$$f \mapsto \int_{x_0}^{x_1} \sqrt{ 1+|f'(x)|^2 } \; \mathrm{d}x$$

Vector scalar product

Given any vector $$\vec{x}$$ in a vector space X, the scalar product with another vector $$\vec{y},$$ denoted $$\vec{x}\cdot\vec{y}$$ or $$\langle \vec{x},\vec{y} \rangle$$, is a scalar. The set of vectors such that this product is zero is a vector subspace of X, called the null space or kernel of X.
Functional equation
Main article: Functional equation

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

$$f\left(x+y\right) = f\left(x\right) + f\left(y\right).$$

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.

Linear functional
Optimization (mathematics)
Tensor

References

Rowland, Todd, "Functional" from 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, MR1878556

Mathematics Encyclopedia