- Functional (mathematics)
In

mathematics , a**functional**is traditionally a map from avector space to the field underlying the vector space, which is usually thereal number s. In other words, it is a "function that takes a vector as its argument or input and returns a scalar". Its use goes back to thecalculus of variations where one searches for a function which minimizes a certain functional. A particularly important application inphysics is to 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 vector space into the underlyingscalar field (usually, real or complex numbers). A special kind of such functionals,linear functional s, gives rise to the study ofdual space s.Transformations of functions is a somewhat more general concept, see

operator .**Examples****Duality**Observe that the mapping:$x\_0mapsto\; f(x\_0)$is a function, here $x\_0$ is an argument of a function. At the same time, the mapping of a function to the value of the function at a point:$fmapsto\; f(x\_0)$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 functional s.**Integral**Integral s such as:$fmapsto\; I\; [f]\; =int\_\{Omega\}\; H(f(x),f\text{'}(x),ldots);mu(mbox\{d\}x)$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"::$fmapstoint\_\{x\_0\}^\{x\_1\}f(x)dx$

*Lp norm of functions::$fmapsto\; left(int|f|^pdx\; ight)^\{1/p\}$

* thearclength of a curve in "n"-dimensional space::$fmapsto\; int\_\{x\_0\}^\{x\_1\}sqrt\{1+|f\text{'}(x)|^2\}dx$**Vector scalar product**Given any vector

**x**in a vector space X, thescalar 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**.**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").

**Functional derivative and functional integration**Functional derivative s are used inLagrangian 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 alsocalculus of variations .Richard Feynman used functional integrals as the central idea in his sum over the histories formulation ofquantum mechanics . This usage implies an integral taken over somefunction space .**References***Eric W. Weisstein et al. " [

*http://mathworld.wolfram.com/Functional.html Functional*] ." From "MathWorld--A Wolfram Web Resource".

* Serge Lang, "Algebra", pub. John Wiley (1965), pp88-91. Library of Congress ref 65-23677.

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