Chapter 3.1, Problem 12E

Let $R^{+}$ denote the set of positive real numbers. Define the operation of scalar multiplication, denoted $\circ$ , by $\alpha \circ x=x^{\alpha }$ for each $x\in R^{+}$ and for any real number $\alpha$ . Define the operation of addition, denoted $\oplus$ , by $x\oplus y=x\cdot y$ for all $x,y\in R^{+}$ Thus, for this system, the scalar product of −3 times $\frac{1}{2}$ is given by

$-3\circ \frac{1}{2}={\left(\frac{1}{2}\right)}^{-3}=8$

and the sum of 2 and 5 is given by $2\oplus 5=2\cdot 5=10$ Is ${\text{R}}^{+}$ a vector space with these operations? Prove your answer.