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Functional


A functional is a real-valued function on a vector space V, usually of functions. For example, the energy functional on the unit disk D assigns a number to any differentiable function f:D->R,

 E(f)=int_D||del f||^2dA.

For the functional to be continuous, it is necessary for the vector space V of functions to have an appropriate topology. The widespread use of functionals in applications, such as the calculus of variations, gave rise to functional analysis.

The reason the term "functional" is used is because V can be a space of functions, e.g.,

 V={f:[0,1]->R:f is continuous}

in which case T(f)=f(0) is a linear functional on V.


See also

Calculus of Variations, Coercive Functional, Current, Euler-Lagrange Differential Equation, Functional Analysis, Functional Derivative, Functional Equation, Generalized Function, Laplacian, Lax-Milgram Theorem, Linear Functional, Operator, Riesz Representation Theorem, Vector Space

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Functional." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Functional.html

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