Definition:Functional/Real

Definition
Let $S$ be a set of mappings.

Let $J: S \to \R$ be a mapping:
 * $\forall y \in S: \exists x \in \R: J \sqbrk y = x$

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

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