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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 ;-))?

Make42
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    I would say the latter. – Sahiba Arora Jun 22 '20 at 15:04
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    Neither sounds wrong. Also, Wiki uses "over" for related topics like https://en.wikipedia.org/wiki/Algebra_over_a_field – VIVID Jun 22 '20 at 15:11
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    @Make42 "an Euclidean space" is also incorrect. It should definitely by "a Euclidean space." – Badam Baplan Jun 22 '20 at 16:08
  • @BadamBaplan: Uh... wow, interesting mistake: I know the rules, but I made the mistake of pronouncing Euclidean wrongly! Thanks! – Make42 Jun 22 '20 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.). – Dave L. Renfro Jun 22 '20 at 17:07
  • @DaveL.Renfro: When you write "be more explicit" - do you mean "more general"? What would be the advantage? – Make42 Jun 22 '20 at 17:48
  • I guess I mean that I don't know what you mean by "Euclidean space". If this is clear in the context where it appears (and it very well could be), then there is not a problem. For example, do you mean ${\mathbb R}^n$ (which itself could be ambiguous, but usually OK in most situations), or do you mean something like "a finite dimensional real inner product space" (in which the elements might not be ordered $n$-tuples of real numbers, although it would be isomorphic to this). If you don't know the difference, I'm guessing you probably intend ${\mathbb R}^n$ (specifying values of $n).$ – Dave L. Renfro Jun 22 '20 at 18:36
  • In short, I'm wondering why use the fancy phrase "Euclidean space"? Again, depending on context this might be fine, but in reading your sentence in isolation from anything else (i.e. the context in which it appears), it sounds to me like you COULD BE unnecessarily using a fancy term for reasons of impressing readers. As I said, this might not be the case, and I'm pointing this out based ONLY on what you've written. I don't know whether this is the case or not (perhaps done unconsciously), and so I thought I'd mention it in case you hadn't given it any thought. – Dave L. Renfro Jun 22 '20 at 18:41
  • @DaveL.Renfro: According to https://en.wikipedia.org/wiki/Euclidean_space#Technical_definition, "finite-dimensional inner product space over the real numbers" is the definition of "Euclidean space". Anyhow, I mean this and also like to keep the properties of being able to calculate an angle and a distance. Simply $\mathbb R^n$ is - as far as I understand - only a set of $n$-tuples, but not necessarily part of an algebraic structure. $\mathbb R^n$ can be the underlying set of a ... space, but it is not the space itself. In my question I abuse terminology. I get this from the discussions: – Make42 Jun 23 '20 at 10:47
  • https://math.stackexchange.com/questions/3706151/calling-a-space-a-set-abuse-of-terminology https://math.stackexchange.com/questions/3720554/vectors-vector-space-and-their-underlying-sets-getting-it-correct-and-elegant https://math.stackexchange.com/questions/3720497/vector-space-it-is-a-not-a-set-is-it https://math.stackexchange.com/questions/3705759/name-of-a-tuple-of-a-set-and-a-function-x-f-akin-to-topological-space-or https://math.stackexchange.com/questions/3705676/name-of-the-set-that-forms-a-topological-space-with-a-topology – Make42 Jun 23 '20 at 10:48
  • In my field (machine learning), the term "Euclidean space" is the opposite of impressing readers. This is the standard "thing" everyone builds on. In fact "a finite dimensional real inner product space" would be a case of impressing readers. The "Euclidean space" has so many preferable properties that it makes it much easier to develop algorithms for it than for more general spaces. (So, it is less impressive to develop for an Euclidean space.) Either way, I am not about impressing readers, but getting it right, so I am glad that you are entering the discussion. – Make42 Jun 23 '20 at 10:55
  • "over" doesn't sound like good idiomatic mathematical English to me ("Algebra over a field" is a different kind of statement). Say "on" in your example and no one will complain. – Rob Arthan Jun 26 '20 at 22:18

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