Usually vector space is well defined by addition and multiplication rules, either between 'scalars' and vectors, or between vectors themselves.
How can I prove that the following addition and multiplication between members form a vector space:
$$u + v = uv \space ; \space \space u,v \in R^{+}.$$ $$a \cdot u = u^{a} \space ; \space \space u \in R^{+} \space \space \space a \in R.$$
When they clearly break the rules?
You have a list of properties to check. So, check them. For instance, addition is commutative because$$u+v=uv=vu=v+u.$$And we have $(a+b)u=au+bu$ (if $a,b\in\Bbb R$ and $u\in\Bbb R_+$) because$$(a+b)u=u^{a+b}=u^au^b=u^a+u^b=au+bv.$$And so on…