\begin{definition}[Definition:Functional/Real]
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.
{{research|Investigate whether the set of mappings $S$ actually need to be real functions or real-valued functions themselves, or whether their codomains are completely arbitrary.
The word 'functional' is usually ambiguous and can mean many things. (Function valued function, function with domain a set of functions, function on a Banach space, element of some kind of (continuous) dual space, ...) I propose to delete. --Wandynsky (talk) 01:36, 31 July 2021 (UTC)
NO to deletion. This page is directly referenced in a specific source work. User:Julius invited to comment.
This is a standard notion in Calculus of Variations, so I am against deletion. Instead, we should write up a disambiguation page. I would do it myself, but I simply have no time nor knowledge to account for every possible context where functional is introduced. In general, when defining terms, I check what disambiguation pages we have here. If there is such a page, then I transclude my definition. If there is no such a thing, I claim the general name, and modify it only if I find a different version in a different book, or someone else needs it in a different context. The tree grows from the ground.
I will make a note to provide some sort of disambiguation page as soon as we have some various definitions of Functional to disambiguate them on. --prime mover (talk) 13:46, 31 July 2021 (UTC)}}
\end{definition}