Just to verify, vector addition is defined as: ?

Let's go through some of the axioms and see if you can do the rest.

1.

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).

__________________________________________________ ________

3.

Pick two arbitrary vectors ap(a) and bp(b). Then:

__________________________________________________ ________

4.

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)

etc. etc.

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.