[ogogmad]

The term "function" sadly means different things in different contexts. I feel like this whole thread is evidence of a need for reform in maths education from calculus up. I wouldn't be surprised if you understood all of this, but I'm worried about students encountering this for the first time.

[skhunted]

Don’t know if you are a mathematician or not but mathematically speaking “function” has a definition that is valid in all mathematical contexts. Functional clearly meets the criteria to be a function since being a function is part of the definition of being a functional.

[ogogmad]

The situation is worse than I thought. The term "function", as used in foundations of mathematics, includes functionals as a special case. By contrast, the term "function", as used in mathematical analysis, explicitly excludes functionals. The two definitions of the word "function" are both common, and directly contradict one another.

[skhunted]

By contrast, the term "function", as used in mathematical analysis, explicitly excludes functionals. The two definitions of the word "function" are both common, and directly contradict each other.

This is incorrect. In mathematics there is a single definition of function. There is no conflict or contradiction. In all cases a function is a subset of the cross product of two spaces that satisfies a certain condition.

What changes from subject to subject is what the underlying spaces of interest are.

[ogogmad]

> What changes from subject to subject is what the underlying spaces of interest are.

I'm not sure I understand what you mean here. I need some clarification. How does this have any bearing on whether functionals count as functions or not? What is the "underlying spaces of interest" in this example?

In some trivial way, every mathematical object can be seen as a function. You can replace sets in axiomatic set theory with functions.

[skhunted]

Everything I wrote was assuming set theory as the foundations for mathematics and applies only to that setup. At any rate a functional is function since the definition starts with: a functional is a function from…

Some books will say: a functional is a linear map….

Note that a linear map is a function.

[ogogmad]

You genuinely don't know what you're talking about. The word "function" means different things in different areas. So does the word "map" or "mapping". In analysis, what you personally call a "function" instead falls under the term "mapping". In foundations - which is a different area with incompatible terminology - the terms "mapping" and "function" are defined to mean the same thing.

This situation is a consequence of how mathematicians haven't always been sure how to define certain concepts. See "generating function" for yet another usage of the word "function" that's in direct contradiction with the last two. So that's three incompatible usages of the term "function". All this terminology goes back to the 1700s when mathematics was done without the rigour it has today.

I find it aggravating how you're so confidently wrong. I hope it's not on purpose.

[edit] [edit 2: Removed insults]