Let $\mathbb{R}+$ be the set of all positive real numbers. Define the operations of addition and scalar multiplication as follows:
$u + v = u.v$ $\forall u,v \in \mathbb{R}+$
$au = u^a$ $\forall u \in \mathbb{R}+$ and real scalar $a$.
Prove that $\mathbb{R}+$ is a real vector space.
I am able to verify all the axioms for it to be vector space except inverse element axiom. Is question correct? Should it be defined over $\mathbb{R}$ instead of $\mathbb{R}+$?
The inverse of $u\in\mathbb{R}^+$ will be $\frac1u$ in that vector space. Note that the zero element of that vector space is $1$, since $(\forall u\in\mathbb{R}^+):1.u=u.1=u$. So, since $u.\frac1u=\frac1u.u=1$, the inverse of $u$ is $\frac1u$.