Just to verify, vector addition is defined as: ?
Let's go through some of the axioms and see if you can do the rest.
Let's choose two arbitrary vectors, ap(a) and bp(b). Then, as it is in the form xp(x) (x = a + b) in this case.
2. Given scalar k, , then as well.
Again, pick an arbitrary vector ap(a). Then (imagine x = ka).
Pick two arbitrary vectors ap(a) and bp(b). Then:
Pick arbitrary vector ap(a). Then, (imagine x = (c+k)a)
5. There exists a zero vector, , such that
Again, choose arbitrary vector ap(a). Then:
(just going by definition)
As you can see, verifying some of these are pretty trivial. Now, it looks like the set is indeed a vector space but I may be wrong. Go through all the axioms and see if they are all satisfied.