Definition:Functional/Real
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Further research is required in order to fill out the details, namely:
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.
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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.
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.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more. To discuss this point in more detail, feel free to use the talk page.
If you have been able to complete the research such that the question has been answered adequately, then you can 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