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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.

John
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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).

Ian
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  • I have clarified the aspect I wish to distinguish. Please take another look If you can! Thanks! – John Jul 22 '15 at 11:55
  • @John The latter kind of object is called a higher order function. Some higher order functions are functionals, if the space of functions is made into a vector space. But not all of them are. – Ian Jul 22 '15 at 11:56
  • I guess my confusion comes from my computer science background. I'm trying to read a book on domain theory, obviously more aimed at mathematicians. I believe they use the word "functional" as meaning "higher order" (i.e. a function which has a function as input) but I just wish to clear this up. Thanks again. – John Jul 22 '15 at 12:01
  • @John I may have a bias from constant exposure to functional analysis. Can you give an example (1-2 sentences) of this usage? It may be a correct usage and the term may just be ambiguous when speaking to mixed company about mixed topics. – Ian Jul 22 '15 at 12:25
  • I think, from the book, its clear they mean "a function which has a function as input". A short sentence from the book is "a program gives rise to an explicitly defined functional F(f;a1,..,an)." – John Jul 22 '15 at 12:56
  • @John Then I agree, in context it seems to make sense. I wouldn't recommend using the term this way among mathematicians, or at least among analysts. – Ian Jul 22 '15 at 13:17