Why is a Zero Vector Space a Vector Space? The first example in my Linear Algebra textbook states this is so by stating the example satisfies axiom $4$, which says there is zero vector, there is an object $u$ in which $0 + u = u + 0 = u$, where $0$ is the zero vector and $u$ is an object or a set of real number.

Is there not already a violation in axiom 1 where the zero space is not closed under addition because the vector space is missing another object v?
The zero set is closed under addition: $0+0=0$. It is closed under scalar multiplication: $\lambda\cdot 0=0$. It is non-empty. Hence it is a vector space, and in fact the only one of dimension $0$ (up to isomorphism and once the field is fixed).
The zero set satisfies the two properties of a subspace.

i) Closed under addition. $0+0=0$

ii) Closed under scalar multiplication $\lambda(0)=0$. 

My previous edit was naively wrong.