This is a fine example of irony.

Let V be a vector space over the reals and L a functional. Let v be a particular element of V. L(v) is a real number. It is a single value. L(v) can't be 1.2 and also 3.4. Thus L is a function.

A function is simply a subset of the product of two sets with the property that if (a,b) and (a, c) are in this subset then b=c.

Can you find a functional that does not meet this criterion? If so then you have an object such that L maps v to a and also maps v to c with a and c being different elements.

Find me a linear map that does not meet the definition of function. Give an example of a functional in which the functional takes a given input to more than one element of the target set.

I think you are not a mathematician and you also don't appear to understand that a word can have different meanings based on context. "generating function" isn't the same thing as "function". Notice that generating is paired with function in the first phrase.

Example: Jellyfish is not a jelly and not a fish. Biologists have got it all wrong!