A Vector Space is a Set - Axiom or Derivation? I understand that structures with the properties of the real and complex numbers can be defined and derived from the axioms of ZFC set theory. 
But can a structure with the properties of a (possibly uncountably infinite dimensional) vector space be derived from ZFC or is it axiomatic statement that a vector space is a set (and that therefore the axioms of set theory apply to it) ?
I think the question becomes relevant in the proof that every vector space has a basis (from Zorn's lemma = axiom of choice). It's not just a matter of whether one accepts the axiom of choice in set theory, but how one gets to believe that a vector space complies with set theory axioms.
 A: This is just a continuation from the definition of $\Bbb R$ as a set. The vector space $\Bbb R^n$, for example, is just the set of tuples with pointwise addition. And trivially, if $V$ is a real vector space whose basis is $B$, then $V$ can be thought of as the set of all functions from $B$ into $\Bbb R$ such that only finitely many elements are mapped to a non-zero element.
The axioms of set theory do not stop, or even begin to stop, at the real numbers or the complex numbers. They go well beyond your wildest dreams1. And then some.

Footnotes:


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*At least if you dream about mathematics and set theory.

A: One can define 
Definition. A vector space over a field $F$ is an abelian group $V$ together with an action $F\times V\to V$ of the field on it.
Definition. An abelian group is a group where the compostion is abelian.
Definition. A group is a set with a binary operation satisfying ...
Ultimately, this makes a vector space essentially a set - that is: A vector space is a tuple of several things and structures, the most important one being the set of vectors of the space.
A: There is a general construction beginning with the natural numbers.
I prefer to go from the non-zero natural numbers to the positive rationals.  Ordered pairs of naturals inherit the order relation of the naturals through the arithmetic of ratios.  Hence, 
these ordered pairs can be collected into equivalence classes and called positive rational numbers.
By a similar process of using ordered pairs, one goes from the positive rationals to the full set of rationals.  In this case, ordered pairs are interpreted by signed differences.  If an ordered pair has the same positive rational for both elements, it will be in the equivalence class designated as 0.
From the rationals, you perform the Dedekind cut construction.  By the power set operation applied to omega in the set universe, you know that there is set in the universe of the proper cardinality.  The set of all countable sequences of natural numbers is uncountable.  If I recall correctly, it referred to as the Baire space.  It relates to the order relations of the Dedekind cuts through the arithmetic of continued fractions.  So, you have a set in the universe corresponding to your construction.
By the constructions thus far, you already have many of your needed operations.  You must consider anything more you may need to satisfy the definitions of a vector space or an inner product space on your construction.  This is all algebraic.  And, this is also where you accommodate dimensionality.
To obtain a Euclidean manifold, you must use a Cartesian product of your construction and an individual copy of your construction.  An affine space (vector space) or a Euclidean point space (inner product space) is obtained by a mapping from the Cartesian product into 
the individual copy.  The Cartesian product in this case is "just a set".  The individual copy carries the vector space structure or the inner product structure.
As a disclaimer, let me say that serious mathematicians will see every mistake I have made in describing this.  But, I think they would concede that this description has met the general sense of how such a construction ought to proceed.
A: Vector spaces are sets, Cartesian products of fields, and therefore by definition rely on the set axioms.
An other question is how well vector spaces model the real space that surround us all. In the real space there are no dots or lines. In real space there are only relative distances. The real space isn't isomorphic with $\mathbb R^n$ at all. And is no set either.
