# Vector Space — Confusion

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 $\mathbb{R}+$?

• Welcome to MSE. It is in your best interest that you use MathJax. – José Carlos Santos Jun 18 '18 at 13:41
• – lhf Jun 18 '18 at 13:44
• Try to show where you missed the demonstration for the inverse element. – Martigan Jun 18 '18 at 13:44

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 $\frac1u$.
• @Pankaj You seem to forget that the zero vector here is the number $1$. – José Carlos Santos Jun 19 '18 at 9:54
• @Pankaj As I wrote in my answer, the additive inverse of $u$ is $\frac1u$. – José Carlos Santos Jun 19 '18 at 9:56