# Function defined "over" a set or "on" a set?

Is the sentence

Given an Euclidean space $$X$$, over which a function $$f: X → \mathbb R$$ is defined ...

correct, or do I write

Given an Euclidean space $$X$$, on which a function $$f: X → \mathbb R$$ is defined ...

in my publication? With

Since a function is defined on its entire domain, ...

https://en.wikipedia.org/wiki/Domain_of_a_function seems to favour the second, while I saw the first in quite a few of places at math.stackexchange.com, e.g. What is the measure of functions defined over $\overline{\mathbb{Q}}$.

EDIT: https://english.stackexchange.com/questions/117234/function-defined-on-over-the-set-a suggests that it would be "on" if $$f$$ is defined on arbitrary set and "over" if the function is defined over sort of (algebraic) structure (or strictly speaking over the underlying set), where the structure is somewhat "respected" by the function. So it might be be

Given an Euclidean space $$X$$, over which a function $$f: X → \mathbb R$$ is defined ...

Given a set $$X$$, on which a function $$f: X → \mathbb R$$ is defined ...

What is to make of this (as mathematicians, instead of linguists ;-))?

• I would say the latter. Jun 22, 2020 at 15:04
• Neither sounds wrong. Also, Wiki uses "over" for related topics like en.wikipedia.org/wiki/Algebra_over_a_field Jun 22, 2020 at 15:11
• @Make42 "an Euclidean space" is also incorrect. It should definitely by "a Euclidean space." Jun 22, 2020 at 16:08
• @BadamBaplan: Uh... wow, interesting mistake: I know the rules, but I made the mistake of pronouncing Euclidean wrongly! Thanks! Jun 22, 2020 at 16:32
• I think "on" sounds better (at least, I'm quite certain I've heard and read the "on" version a lot more often than the "over" version), but for what you're writing I would not use either, since the notation $f:X \rightarrow {\mathbb R}$ already incorporates specification of the domain. Maybe something like: "Let $X$ be a Euclidean space and $f: X \rightarrow {\mathbb R}$ be a function. Then $\ldots$" Maybe also be more explicit than "Euclidean space" (e.g. a finite-dimensional inner product space not necessarily ${\mathbb R}^n$, an affine space without a designated zero element, etc.). Jun 22, 2020 at 17:07