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A functional is a mapping:

whose domain is a set of mappings
whose codomain is another set of mappings.

Real Functional

Let $S$ be a set of mappings.

Let $J: S \to \R$ be a mapping from $S$ to the real numbers $\R$:

$\forall y \in S: \exists x \in \R: J \sqbrk y = x$

Then $J: S \to \R$ is known as a (real) functional, denoted by $J \sqbrk y$.

That is, a (real) functional is a real-valued function whose arguments are themselves mappings.


Differential Operator

The differential operator is an example of a functional.

Definite Integral

A definite integral is an example of a functional whose image set is a set of numbers.

Also see

  • Results about functionals can be found here.