# Definition:Functional/Real

## Definition

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.

Further research is required in order to fill out the details.In particular: 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)You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Research}}` from the code. |

## 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.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (next): $\S 1.1$: Functionals. Some Simple Variational Problems