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At wikipedia and here at math.stackexchange.com it is often said that a vector space is a set with some properties and then one goes on to refer to the set as the vector space. I am guessing that this is "just" an abuse of terminology and notation, right?

Shouldn't the tuple $(X, S, +, *)$ be the vector space, where $X$ is a set, which elements we can add up using $+$ (and receive an element of $X$), and which we can multiply with elements of the set $S$ (and receive an element of $X$) using $*$? Often it is said that $X$ is the vector space, but that is strictly speaking wrong isn't it? I am asking, so that I may understand the mathematical concepts better.

Bonus question: $X$ would be the underlying set of the vector space $(X, S, +, *)$, I guess. Could I say that, if $X=\mathbb R^n$, $X$ is the Euclidean spaced set or do I have to go route of writing in my publications that "$X$ is the underlying set of an Euclidean space"?

Background: I submitted a paper and, even though it was ultimately accepted, one of the reviewers said that my math is not precise enough. I think (s)he might be right... especially with this topic.

Make42
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    A vector space is a Mathematical structure i.e. a set plus one or more operations defined on it. – Mauro ALLEGRANZA Jun 15 '20 at 11:13
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    See also Space (mathematics) – Mauro ALLEGRANZA Jun 15 '20 at 11:15
  • Almost everything in maths is a set. Don't forget that sets can be members of sets. – badjohn Jun 15 '20 at 11:15
  • Using the same letter for a vector space and the underlying set may technically be an abuse, but it’s very common – J. W. Tanner Jun 15 '20 at 11:18
  • See also another previous post – Mauro ALLEGRANZA Jun 15 '20 at 11:25
  • I think this question was pretty messed up by my edit. I tried to re-edit but realized, now my question is kind of very different, so I ask a new one. – Make42 Jun 15 '20 at 11:59
  • You are correct that a vector space is more than just its underlying set. However, the conflation of a vector space with its underlying set is universally accepted - provided that it's clear from context what the additional structure is. (And indeed this is true for structures in general - groups, rings, topological spaces, etc.) I suspect this is not the point the reviewer was objecting to. Ideally their objection should point to some specific points. – Noah Schweber Jun 15 '20 at 12:21
  • I honestly struggle to imagine a situation where speaking of "an element of a vector space" would lead to confusion. Technically, yes, the elements belong to the associated set, not the space itself, but nobody makes that distinction in practice, unless they are writing a paper where different spaces over the same set are involved. If the reviewer was criticising this point, then they were needlesly pedantic. – R_B Jun 15 '20 at 12:41

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