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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.