Let V be the abelian group of positive real numbers for multiplication.Define scaler multiplication in V by Let V be the abelian group of positive real numbers for multiplication.Define scaler multiplication in V by ax=x^a ,a real,and x \in V .Show that V is vector space over reals. i) I know that V is an abelian group with respect to addition,but here it is mentioned that V is an abelian group for multiplication!!! Why? ii) If I do a(x + y) \neq (x+y)^a, then how it is a vector space over real!

Your vector 'addition' is defined as multiplication, so in your case, showing $a(x+y) = ax + ay$ (as vectors) means showing that $(xy)^a = x^ay^a$, which is certainly true for the positive reals. It's important to remember that you have to prove the scalar axioms in terms of how your vector addition is defined, which in your case is multiplication of positive reals.

As for why the positive reals are a group under multiplication, it should be easy to check that associativity holds, inverses exist, that there is an identity ($1$), and that it's closed; you can't get a non-positive real by multiplying two positive reals. If you check these axioms one by one, you will probably be able to convince yourself that it is a group.