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