TITLE: How do I conceptualize a vector space
QUESTION [5 upvotes]: I have read a number of articles and posts about vector spaces I am still left unconfident that my conceptual model of a vector space is correct.
Here is an example to illustrate how I am thinking about them currently. To simplify things lets put a constraint on our vector space and only focus on a subset of the Real numbers. Imagine a 2d space with points (-5,5) (5,5) (-5,-5) (5,-5) that create a bounding area for our vector space. We can assume all the axioms of a vector space apply to this space (essentially scalar multiplication and vector addition).
Would the vector space be the set of all possible vectors that can fit in that box? Or would it be the space in which the vectors live (the 2d box)? Wikipedia describes a vector space as "a collection of objects, called vectors", which makes me think it is the former.

REPLY [2 votes]: I think the essence of this question is not really about the specifics of vector space axioms, but more generally, whether the "vector space" is some kind of container (like some background or stage) or the collection of all things that can be put into said container (set of vectors).
Such philosophical questions are rarely discussed. The short answer is: things in math are whatever people have defined them to be. How to best think about them intuitively is a separate question.
Over the last century or so, set theory has become the basis of most math. Even for primary school math. For example, in school (depending on country etc.) we learn that geometric shapes, like circles "are" sets of points.
Vector spaces are sets with operations defined on them, fulfilling certain axioms. However, the distinction often isn't clearly made between the set and the vector space, even (or especially) in academic texts. The vector space may be denoted with the same letter as the underlying set. But this is just an abuse of notation to reduce clutter for experienced readers (although it confuses beginners who like to see things written out explicitly and less ambiguously).
I think in practice, most people intuitively work with the "vector space as background stage" mental model, though, because it's hard to visualize or imagine all the elements (vectors) populating the space at the same time. I usually imagine the vector space as just sitting there, and focus my attention on a few particular vectors under analysis as if they were floating in there in an empty space. But again, this is just how my monkey brain deals with it, this has no bearing on what a vector space mathematically is.