Consider $V = \{x \in \mathbb{R}: x > 0\}$. Define addition by $x\oplus y := xy+1$ and scalar multiplication by $r\otimes x := r^2x$. Prove if V is a vector space using the vector space axioms and provide a counterexample if not.
I know that $V$ is closed under vector addition and scalar multiplication. I also know that it is commutative and associative.
How do I prove axioms 4 and 5. Determining if a zero element exists and if there is an inverse satisfying these properties. If they do not exist, please help with a counterexample.
Let's say $y$ is the zero element. Then for all $x$, $xy+1=x$. But we can solve for $y$, which gives $y=\frac{x-1}x$ – not a constant. Therefore $V$ has no zero element, and thus no inverse element.