# Definition:Functional/Real

Jump to navigation
Jump to search

## 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.

## 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