Is R^+ a Vector Space with Non-Standard Operations?

[mpm]

I have a homework problem that I can't figure out and there is nothing in the book that helps me out. I was hoping someone could shed some light.

Let R^+ denote the set of positive real numbers. Define the operation of scalar muplication, denoted * (dot) by,

a*x = x^a

for each X (episilon) R^+ and for any real number a. Define the operation of addition, denoted +, by

x + y = x * y for all x, y (Epsilon)R^+

Thus for this system the scalar product of -3 times 1/2 is given by

- 3 * 1/2 = (1/2)^-3 = 8

and the sume of 2 and 5 is given by

2 + 5 = 2 * 5 = 10

Is R^+ a vector space with these operations? Prove your answer.

The plus should be a plus with a circle around it but I couldn't figure out how to put it in there. I am also not sure how to make the epsilon either.

Any help would be greatly appreciated.

mpm

 

[AKG]

This is an epsilon ($\epsilon$), you're looking for a different symbol, the "is a member or element of" relation ($\in$). So you have:

$$(\mathbb{R}^+, \oplus, \otimes)$$

with the following definitions, for all x, y in R⁺ and all scalars (reals) $\lambda$:

$$x \oplus y = x \times y$$

$$\lambda \otimes x = x^{\lambda}$$

Do you know the definition of a vector space? Basically, all you have to do is check that the operations are well-defined, and then show that they satisfy all the properties (like commutativity of addition, associativity of scalar multiplication, existence of identities, etc.).