What is the difference between function and functional in mathematics?

Answer 1

In order to determine a difference between two aspects in question, the two aspects need to be of comparable nature. This is why this question can not officially be answered due to the simple fact that a functional and a function are two objects in mathematics that are simply incomparable.

The reason a function can not be compared to a functional is because a functional is in fact a class of functions that, by definition, map the elements of a vector space into the field that underlies the particular vector space, which will be either the real or complex numbers more than often. What is interesting is the fact that even though the class of functions is distinguishable from the class of functionals, the very notion of difference becomes nonsense because one class is contained within the other. This would be the same as considering the difference between Americans and humans.

However, the answer to how one is distinguishable from the other simply comes from their definitions. As a final interesting note, notice that though the objects themselves were incomparable, a distinguishment could be made not by comparing the objects themselves, but through comparison associated with the concept of each object. These are in fact quite interesting topics in the philosophy of mathematics and logic, but for now, I believe the main question at hand has been dealt with.