Q:

General meaning of the term "Functional"

In the most general is a "functional" simply a function which can accept a function as input?
So, is it natural to describe:
$f: \mathbb{N} \rightarrow \mathbb{N}$
as a function. Whereas it is more natural to describe:
$F: (\mathbb{N} \rightarrow \mathbb{N}) \rightarrow \mathbb{N} $
As a functional (although it is obviously still a function).
The context is a book on Domain Theory, where they describe:
$$GCD(h;a,b) :=  \begin{cases}b & \text{if } rem(a,b) = 0 \\ h(b,rem(a,b)) & \text{otherwise} \\\end{cases}$$
as a functional.

A:

A functional is a function from a vector space to its underlying field. Often the vector space is a function space, in which case the argument is indeed a function, but this isn't always the case. In general a function which takes a function as an argument is called a higher order function (although this term is really from computer science, not mathematics).

