On the first page of the classical book "Ordinary Differential Equations" by Jack Hale (Revised Edition, 1980) there is the following definition:

> 
> An *abstract linear vector space* (or *linear space*) $\mathcal{X}$ over $\mathbb{R}$ is a collection of elements $\{x,y,\ldots\}$ such that for each $x,y \in \mathcal{X}$, the sum $x+y$ is defined, $x+y \in \mathcal{X}$, $x+y=y+x$ and there is an element $0 \in \mathcal{X}$ such that $x+0=x$ for all $x \in \mathcal{X}$. Also, for any number $a,b \in\mathbb{R}$, scalar multiplication $ax$ is defined, $ax \in \mathcal{X}$ and $1 \cdot x = x$, $(ab)x=a(bx)=b(ax)$, $(a+b)x=ax+bx$ for all $x,y \in \mathcal{X}$.
> 
> 
> 

The terminology *linear* vector space is the same as vector space (i.e., without the adjective *linear*)? I am asking this because a classical axiom of vector spaces is missing here: given an $x \in \mathcal{X}$ there is an element $z \in \mathcal{X}$ such that $x+z=0$, where the element $0$ was defined above.

**Question improvement**: with respect to the definition of *vector space*, more axioms seem to be missing too, namely the associativity under $+$ and the scalar distributivity as $a(x+y) = ax + ay$. This was mentioned by more than one comment/post of contributors. 

Why is that?