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 ${\displaystyle V}$ into its field of scalars, i.e., it refers to an element of the dual space ${\displaystyle V^{*}}$.
• In mathematical analysis, more generally and historically, it refers to a mapping from a space ${\displaystyle X}$ into the real numbers, or sometimes into the complex numbers, for the purpose of establishing a calculus-like structure on ${\displaystyle X}$. Depending on the author, such mappings may or may not be assumed to be linear, or to be defined on the whole space ${\displaystyle X}$.
• In computer science, it is synonymous with higher-order functions, i.e. functions that take functions as arguments or return them.

The arc length functional has as its domain the vector space of rectifiable curves (a subspace of ${\displaystyle C([0,1],\mathbb {R} ^{3})}$), and outputs a real scalar. This is an example of a non-linear functional.

The Riemann integral is a linear functional on the vector space of Riemann-integrable functions from a to b, where a,b ∈ ${\displaystyle \mathbb {R} }$.

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 ${\displaystyle X}$ 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 ${\displaystyle X}$ 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

${\displaystyle x_{0}\mapsto f(x_{0})}$

is a function, where x₀ 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

${\displaystyle f\mapsto f(x_{0})}$

is a functional; here, x₀ 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

Integrals such as

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

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

• the area underneath the graph of a positive function ${\displaystyle f}$

${\displaystyle f\mapsto \int _{x_{0}}^{x_{1}}f(x)\;\mathrm {d} x}$

• L^p norm of a function on a set ${\displaystyle E}$

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

• the arclength of a curve in 2-dimensional Euclidean space

${\displaystyle f\mapsto \int _{x_{0}}^{x_{1}}{\sqrt {1+|f'(x)|^{2}}}\;\mathrm {d} x}$

Inner product spaces

Given an inner product space ${\displaystyle X}$, and a fixed vector ${\displaystyle {\vec {x}}\in X}$, the map defined by ${\displaystyle {\vec {y}}\mapsto {\vec {x}}\cdot {\vec {y}}}$ is a linear functional on ${\displaystyle X}$. The set of vectors ${\displaystyle {\vec {y}}}$ such that ${\displaystyle {\vec {x}}\cdot {\vec {y}}}$ is zero is a vector subspace of ${\displaystyle X}$, called the null space or kernel of the functional, or the orthogonal complement of ${\displaystyle {\vec {x}}}$, denoted ${\displaystyle \{{\vec {x}}\}^{\perp }}$.

For example, taking the inner product with a fixed function ${\displaystyle g\in L^{2}([-\pi ,\pi ])}$defines a (linear) functional on the Hilbert space ${\displaystyle L^{2}([-\pi ,\pi ])}$of square integrable functions on ${\displaystyle [-\pi ,\pi ]}$:

${\displaystyle f\mapsto \langle f,g\rangle =\int _{[-\pi ,\pi ]}{\bar {f}}g}$

Locality

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:

${\displaystyle F(y)=\int _{x_{0}}^{x_{1}}y(x)\;\mathrm {d} x}$

is local while

${\displaystyle F(y)={\frac {\int _{x_{0}}^{x_{1}}y(x)\;\mathrm {d} x}{\int _{x_{0}}^{x_{1}}(1+[y(x)]^{2})\;\mathrm {d} x}}}$

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

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

${\displaystyle f(x+y)=f(x)+f(y).}$

Derivative and integration

See also: Calculus of variations

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

• Linear form
• Optimization (mathematics)
• Tensor

References

• "Functional", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• 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