Question

Isomorphism?

Answer 1

You need to clarify:

1. The definitions of your domain/codomain. It's not clear to me what R\* is, or R\*\* for that matter.
2. The structure that the isomorphism is supposed to preserve. Is it a group isomorphism? Something else?

Classic example of a Group Isomorphism:

>f : (**R**, +) → (**R**^(>0), ×)  
>  
>f: x ↦ e^(x)

i.e. The exponential function is an isomorphism between the group formed by the real numbers under addition and the positive real numbers under multiplication. It is both a group homomorphism and a bijection (making it a group isomorphim).

Answer 2

What is R*? Nonnegative real numbers? At any rate, it's not an isomorphism because it's not injective (and not even a homomorphism)

Answer 3

Is it injective? Not sure what R** is supposed to mean though

An example would be f:C -> C, f(z) = complex conjugate of z