I got pretty confused during my attempt to use the set mentioned in the title, namely the set of all vector spaces that use the same set of scalars $S$, as the set of 'vectors' to make a new vector space, $V$. I defined the 'sum' of two 'vectors' (vector spaces), $V_i$ and $V_j$ as their tensor product space $V_i\otimes V_j$, and used the set $S$ as my set of scalars, defining 'multiplication by a scalar' by $cV_p$, $c\in S$ meaning the vector space $V_q$ containing all the vectors in $V_p$ except multiplied by $c$, which should be OK I think since all of the vector spaces in $V$ have a clear meaning to multiplying by an element of $S$ since that's what 'multiplication by a scalar' in those vector spaces means (although given that those vector spaces are closed under this operation, it seems that my scalar multiplication basically means 'do nothing' (which is fine by me)). In the book I'm reading it says the set $V_i\otimes V_j$ is a vector space, and it seems that $\otimes$ does all the things you'd want vector addition to do, and so it seems $V$ satisfies all the axioms needed for it to be a vector space, apart from two: I don't see how it can contain a zero element, nor an additive inverse for each element in $V$.
The reason I'd want to do such a crazy thing is because if I can make this set $V$, the set of all vector spaces using the $S$ for their concept of scalar multiplication, into a vector space, in particular one that uses $S$ as I've described, then surely $V\in V$ !
So is there some way to fix this, either by coming up with some clever vector spaces to be additive inverses and the zero element or perhaps by redefining vector addition? Or is this set $V$ doomed to never be a vector space?