Functional
High school
A functional takes a whole function as its input and returns a number.
Undergraduate (functional analysis)
Consider a vector space, often a space of functions, and its scalars. A functional is a map from the vector space to its scalars.
Examples
Examples
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$f \mapsto f(0)$
Undergraduate
$f \mapsto f(0)$ is a functional, because it takes a function and returns a number.
Non-examples
Things that may be mistaken for examples.
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$f \mapsto f^{\prime}$, on differentiable functions
High school
$f \mapsto f^{\prime}$, on differentiable functions, is not a functional, because its outputs are functions, not numbers.
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$f(x) = x^2$ on $\reals$
High school
$f(x) = x^2$ on $\reals$ is not a functional, because its inputs are numbers, not elements of a space of functions.