Functional (mathematics)

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In mathematics, a functional is traditionally a map from a vector space to the field underlying the vector space, which is usually the real numbers. In other words, it is a function that takes a function as its argument or input and returns a scalar. Its use goes back to the calculus of variations where one searches for a function which minimizes a certain functional. A particularly important application in physics is search for a state of a system which minimizes the energy functional.

In functional analysis, the functional is also used in a broader sense as a mapping from an arbitrary linear vector space into the underlying scalar field (usually, real or complex numbers). A special kind of such functionals, linear functionals, gives rise to the study of dual spaces.

Transformations of functions is a somewhat more general concept, see operator.

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[edit] Examples

[edit] Duality

Observe that the mapping

x_0\mapsto f(x_0)

is a function, here x0 is an argument of a function. 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 x0 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.

[edit] 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)dx
f\mapsto \left(\int|f|^pdx\right)^{1/p}
  • the arclength of a curve in n-dimensional space
f\mapsto \int_{x_0}^{x_1}\sqrt{1+|f'(x)|^2}dx

[edit] Vector scalar product

Given any vector x in a vector space X, the scalar product with another vector y, x.y, is a scalar. The set of vectors such that this product is zero is a vector subspace of X, called the null space of x.

[edit] 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(x+y) = f(x) + f(y).

[edit] 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.

[edit] References

  • Eric W. Weisstein et al. "Functional." From MathWorld--A Wolfram Web Resource.
  • Serge Lang, Algebra, pub. John Wiley (1965), pp88-91. Library of Congress ref 65-23677.