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

For the functional to be continuous, it is necessary for the vector space 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 can be a space of functions, e.g.,

in which case is a linear functional on .

This entry contributed by Todd Rowland

CITE THIS AS:

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