[gunnihinn]

A distribution is not a function. It is a continuous linear functional on a space of functions.

Functions define distributions, but not all distributions are defined that way, like the Dirac delta or integration over a subset.

[skhunted]

A functional is a function.

[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.