# Definition:Functional/Real

## Definition

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.

## Also known as

A functional can be a mapping from a set into other codomains, for example the complex numbers $\C$.

However, such is the prevalence of real functionals that it is commonplace to refer to them just as functionals, and the codomain is then assumed by default to be the real numbers.