Functionals

In one scenario, a functional takes a vector and returns a scalar.

The forensics mathematician in you says that this can be done only by a covector with the same dimension as the vector.

Another author mentions that a functional takes vectors and makes a scalar.

Functionals are linear so the following ( $(v_1, v_2)$ are vectors and $(\alpha, \beta)$ ) is true:

$f(\alpha v_1 + \beta v_2) = \alpha f(v_1) + \beta f(v_2)$

Appendix

Elsewhere, an author explained that a function takes a number and returns a number while a functional takes a function and returns a function.

As an example, Differentiation with respect to ‘x’ takes $x^2$ and returns $2x$ .

$\dfrac {d}{dx} (x^2) = 2x$