We know that the reason why we want to introduce a vector space and work with a vector space is that we want to work with a set whose elements can be added and scaled (or a set whose elements are closed under addition and scalar multiplication). Given this motivation, I'm confused by the definition of a vector space:
Definition. A vector Space over field $F$ is a set $V$ such that:
(i). Two operations are defined: vector addition: $V × V → V$ ; scalar multiplication:
$F × V → V$ .
(ii). The set V and these two operations satisfy 8 axioms.
Note that as addition is defined as a mapping $V\times V\rightarrow V$, and scalar multiplication is defined as mapping $F\times V\rightarrow V$, this means part (i) of the definition already gives us a set that is closed under finite addition and scalar multiplication.
My question: Given that what we want from a vector space is merely that it is a set whose elements can be added and scaled, what is the primary motivation for imposing the 8 axioms (part (ii) of the definition)?
Best Answer
Though we gave a preliminary name 'vector addition' (and a suggestive notation '$+$') to the operation $V\times V\to V$, it is not assumed that it is indeed 'an addition' operation on some known structures.
In itself it can be any two variable function on $V$.
Instead, we assume the basic and most important properties (the axioms) to try to capture what it means to be an 'addition-like operation'.
Similarly for the scalar multiplication.
Note also that $F$ is already assumed to be a field, i.e. to be equipped with (constants named $0$ and $1$), an addition(-like operation), a substraction(-like operation), a multiplication(-like operation), and a division(-like operation) by any nonzero element.