Vector Space Axioms

A (real) vector space is a triple $(V,+,\cdot )$, where $V$ is a set and $(+,\cdot )$ are operations. In particular, $+:V\times V\to V$ is called vector addition, while $\cdot :\mathbb{R}\times V\to V$ is called scalar multiplication. When $u,v\in V$ and $c\in \mathbb{R}$, we denote these operations as $u+v$ and $c\cdot v=cv$. These two operators must satisfy the following properties:

1. Closure of addition For all $u,v\in V$, $u+v\in V$

2. Commutativity of addition For all $u,v\in V$, $u+v=v+u$

3. Associativity of addition For all $u,v,w\in V$, $(u+v)+w=u+(v+w)$

4. Existence of an additive identity There exists an element $0\in V$ such that $u+0=0+u=u$ for all $u\in V$

5. Existence of an additive inverse For all $u\in V$, there exists an element $-u\in V$ such that $u+(-u)=(-u)+u=0$

6. Closure of scalar multiplication For all $u\in V$ and $c\in \mathbb{R}$, $cu\in V$

7. Associativity of scalar multiplication If $u\in V$ and $c,d\in \mathbb{R}$, then $(cd)u=c(du)$

8. Compatibility If $u,v\in V$ and $c,d\in \mathbb{R}$, then $c(u+v)=cu+cv$ and $(c+d)u=cu+du$

9. Action of the multiplicative identity For any $v\in V$, $1v=v$