Functional (mathematics)

The exact definition of the term varies depending on the subfield (and sometimes even the author).

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.

is a space of functions is not mathematically essential, so this older definition is no longer prevalent.

[citation needed] 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.

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 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.

form a special class of functionals.

into a real number, provided that

Examples include Given an inner product space

and a fixed vector

the map defined by

The set of vectors

called the null space or kernel of the functional, or the orthogonal complement of

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.

This occurs commonly when integrals occur separately in the numerator and denominator of an equation such as in calculations of center of mass.

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:

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.

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.