How to determine vector space?

I am taking a linear algebra course, and we are currently learning about vector spaces and subspaces.

On the beginning of the chapter it is said that vector space must "comply" with all of the ten rules to actually be a vector space.

But throughout the rest of book for each example they only prove it for two axioms:

For example - The set {f | f:N->R } of all real-valued functions of one natural number variable is a vector space under the operations : $$(f_1+f_2)(n) = f_1(n) + f_2(n)$$ and $$(r\cdot f)(n) = r\cdot f(n).$$

and they do not test any further. What's with 8 remaining conditions for vector space?

So my question is, how to determine the vector space, with two axioms or all ten of them? And why do they prove it in book only for those two axioms?

$\cdot$ = multiplication

• Vector space or subspace? – mrs Jan 5 '14 at 16:48
• I think you've misunderstood something. The "2 axioms" you mention are (in slightly garbled form) the definition of a linear transformation, not two axioms for a vector space. – hardmath Jan 5 '14 at 16:48
• If $V$ is a known vector space, and $W$ is a non-empty subset of $V$, then $W$ is a vector space if it satisfies the 2 axioms, since the others are automatic. – Gerry Myerson Jan 5 '14 at 16:48
• @hardmath functions on finite sets can be identified with vectors in a finite dimensional space. – user113529 Jan 5 '14 at 16:57
• For a subset of a vector space one only needs to check these closure axioms since any subset inherits the operations of the vector space. – user113529 Jan 5 '14 at 17:02

From your notation, it seems that your 2 axioms are the definitions of the operations that turn the set $V^A$ of all functions from a set $A$ to a vector space $V$ into a vector space.