The set of all positive real numbers, with addition defined by $x + y = x * y$ and scalar multiplication defined by $c * x = x ^ c$. How is this a vector space? Doesn't the axiom stating that $(c+d)u = cu + du$ fail?
example: $u = 2$ (since two is a positive real number, it is in the set). $c = 3$ and $d = 4$ ( these two are just scalars).
$(c+d)u = (3+4)(2) = 2^7$
$cu + du = 2^3 + 2^4$, which does not equal $2^7$, so the axiom fails.
How is this a vector space if an axiom fails? Or does this axiom somehow not fail? Any help is appreciated.