Comment by skhunted

25 days ago

You genuinely don't know what you're talking about. .... I find it aggravating how you're so confidently wrong.

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!

I'll try one last time.

> I think you are not a mathematician

Guess again.

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

You have a problem with reading comprehension. I never said any mathematician was wrong.

Think about namespaces for a moment, like in programming. There are two namespaces here: The analysis namespace and the foundations namespace.

In either of those two namespaces, the word "mapping" means what you're describing: an arbitrary subset F of A×B for which every element of a ∈ A occurs as the first component in a unique element (x,y) ∈ F.

But the term "function" has a different meaning in each of the two namespaces.

The word "function" in the analysis namespace defines it to ONLY EVER be a mapping S -> R or S -> C, where S is a subset of C^n or R^n. The word "function" is not allowed to be used - within this namespace - to denote anything else.

The word "function" in the foundations namespace defines it to be any mapping whatsoever.

Hopefully, now you'll get it.

  • If one has a “thing” that “maps” elements of one set to another that satisfies the condition I previously gave then that thing is a function. Every functional satisfies that definition. Therefore every functional is a function.