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. I want to really be precise and correct here for the camera-ready version.
Preliminaries
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. It was pointed out to me, that calling the underlying set $Y$ of a topological space $(Y,\tau)$ a "topological space" is a common abuse of terminology. In an answer to this question (now deleted) it was pointed out to me, that the same is the case for vector spaces and their underlying set.
So, the tuple $(X, S, +, *)$ would 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 - even if commonly accepted, because "everyone understands what is meant".
The mentioned answer also pointed out that strictly speaking $S$ is not a set, but a field and thus $S$ should actually be substituted with $(S,+,-,*,/,-,^{-1})$, but that basically nobody does that either.
Questions
I was told that I should call $X$ the "underlying set of the vector space". But what then is a vector? It is often set that "a vector is an element of a vector space", but I think it would be more correct to say that "a vector is a element of the underlying set of a vector space". Is that correct or am I removing the right to call such an element "a vector" once I forget the property of vectoriness, when I say "the set of the underlying vector space". I mean "the set of the underlying vector space" is just a set. Maybe I am not forgetting this though, because I said "underlying"... maybe I am just being pedantic even for a math-person.
Ok, let's assume that "a vector is a element of the underlying set of a vector space". Isn't $X$ then a set of vectors (in contrast with the vecor space, which is not a set)? Could I say then that $X$ is a "vector set" instead of saying that it is the the "underlying set of a vector space"? That would be much shorter... But maybe I am also making a mistake here, because this will be confused with a set of any vectors (so, e.g. a subset of $X$) or alternatively confused with the vectors that are used for a "span of a set of vectors".
Let's say my vector space is also an Euclidean space. 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"?
Regarding my publication: In one passage I write "... which can be expressed as vectors in nine-dimensional Euclidean space." which seems wrong to me now. I think it should be "... which can be expressed as vectors in nine-dimensional Euclidean spaced set" if the above is correct.
In another passage I write "the dataset was linearly mapped into a vector space of higher dimension and added with Gaussian noise" which also seems very wrong now. How do I write this correctly?
