# Definition:Functional

## Definition

Let $J: S \to \R$ be a mapping:

- $\forall y \in S: \exists x \in \R: J \sqbrk y = x$

Then $J: S \to \R$ is known as a **functional**, denoted by $J \sqbrk y$.

That is, a **functional** is a real-valued function whose arguments are themselves mappings.

## Sources

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