In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
, it is synonymous with
linear form
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
s, which are linear mapping from a vector space
into its
field of scalars
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ...
(that is, an element of the
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
)
[ "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''."]
* In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
and related fields, it refers more generally to a mapping from a space
into the field of
real or
complex numbers.
[ "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''(α''x'' + β''y'') = α''f''(''x'') β''f''(''y'') where ''x'', ''y'' ∈ ''R'' and α, β are arbitrary numbers."] In functional analysis, the term is a synonym of
linear form
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
;
[ p. 101, §3.92] 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
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, it is synonymous with
higher-order function
In mathematics and computer science, a higher-order function (HOF) is a function that does at least one of the following:
* takes one or more functions as arguments (i.e. a procedural parameter, which is a parameter of a procedure that is itse ...
s, 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 calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. The first concept, which is more modern and abstract, is discussed in detail in a separate article, under the name
linear form
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
. 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
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
, where one searches for a function that minimizes (or maximizes) a given functional. A particularly important application in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
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
is a function, where
is an
argument of a function
In mathematics, an argument of a function is a value provided to obtain the function's result. It is also called an independent variable.
For example, the binary function f(x,y) = x^2 + y^2 has two arguments, x and y, in an ordered pair (x, y). T ...
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
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
.
Provided that
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 functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
s.
Definite integral
Integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
s such as
form a special class of functionals. They map a function
into a real number, provided that
is real-valued. Examples include
* 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
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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 In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
of
denoted
For example, taking the inner product with a fixed function
defines a (linear) functional on the
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
of square integrable functions on
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:
is local while
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.
Functional equations
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 derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on w ...
s are used in
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
. They are derivatives of functionals; that is, they carry information on how a functional changes when the input function changes by a small amount.
Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...
used
functional integrals as the central idea in his
sum over the histories formulation of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. This usage implies an integral taken over some
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
.
See also
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References
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* {{MathWorld, title=Linear functional, urlname=Linear_functional, author=Rowland, Todd
Types of functions