I have defined function A as A: D*E --> F and function B as B: S*T --> R.
Now, I want to define function C which gets an instance of A functions and an instance of B functions as inputs and returns a real number.
What is the correct notation for this function?
Is the following notation correct?
C: A(.)*B(.) --> R
Since this is a mathematics site, let's use mathematical notation.
There is no widely accepted mathematical notation that looks like this:
A: D*E --> F. However, there is a notation that looks like $A: D \times E \to F,$ which says that $A$ is a function to which we give an element of the set $D$ and an element of the set $E,$ upon which $A$ produces an element of the set $F.$Note that in this notation, $D,$ $E,$ and $F$ are sets that may contain many different elements, but $A$ is the name of a particular function. If you want to describe a function that takes other functions as input, it does not usually make sense to use the name $A$ as part of the "type" of the new function, any more than it makes sense to use the name of the number $3$ as part of the "type" of a function. What I mean is, you would not write $f: 3 \times \mathbb R \to \mathbb R.$ Technically, you could write $f: \{3\} \times \mathbb R \to \mathbb R,$ but that's rather silly; it says the first argument to the function must always be $3,$ which kind of defeats the purpose of giving the argument to the function in the first place.
You certainly can have a function that acts on functions. Here is an example: $$ C: (\mathbb R \times \mathbb R \to \mathbb R) \times (\mathbb R \times \mathbb R \to \mathbb R) \to \mathbb R.$$ This says that $C$ is a function that takes two arguments, each of which itself is a functions that takes two real numbers arguments, and $C$ returns a real number as its result. As a more specific example, we could say that $C$ is the function such that $$ C(f,g) = f(0,0) + g(1,1) $$ for any two functions $f: \mathbb R \times \mathbb R \to \mathbb R$ and $g: \mathbb R \times \mathbb R \to \mathbb R.$