Definition:Functional
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Definition
A functional is a mapping:
Real Functional
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.
Examples
Differential Operator
The differential operator is an example of a functional.
Definite Integral
A definite integral is an example of a functional whose image set is a set of numbers.
Also see
- Results about functionals can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): functional
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): functional