- Functional (mathematics)
mathematics, a functional is traditionally a map from a vector spaceto the field underlying the vector space, which is usually the real numbers. 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 the calculus of variationswhere one searches for a function which minimizes a certain functional. A particularly important application in physicsis to search for a state of a system which minimizes the "energy functional".
functional analysis, the functional is also used in a broader sense as a mapping from an arbitrary 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
Observe that the mapping:is a function, here 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 "functional", here 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
Integrals such as: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"::
Lp normof functions::
arclengthof a curve in "n"-dimensional space::
Vector scalar product
Given any vector x in a vector space X, the
scalar productwith 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.
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 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 Feynmanused 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.
*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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