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.

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    $\begingroup$ check cu + du you forgot a step $\endgroup$ – T. Fo Jan 11 '19 at 14:13
  • $\begingroup$ It's always a bad idea to abuse notation like this...here you, or your source, is using $x+y$ (and $xy$) in two different ways. Even the core definition seems to use $x*y$ in two different ways. Better to introduce new notation for the new operation, otherwise confusion is nearly certain. $\endgroup$ – lulu Jan 11 '19 at 14:15
  • $\begingroup$ But $cu$ and $du$ are vectors and vectors sum is $x * y$. This means that $cu+du=2^3 * 2^4=2^7$. $\endgroup$ – Mauro ALLEGRANZA Jan 11 '19 at 14:15

It definitely is a vector space. You seem to be getting confused about the two $+$ and two $*$ operations, as well as the fact that vectors and scalars have some overlap. It would be better to give the operations their own names:

\begin{align*} u \oplus v &= uv \\ \lambda \odot u &= u^\lambda. \end{align*}

The distributivity law that you're trying to verify is as follows:

$$(\lambda + \mu) \odot u = (\lambda \odot u) \oplus (\mu \odot u).$$

Please note the scalar $+$ and the vector $\oplus$, and where they belong. Don't forget that $\lambda$ and $\mu$ are scalars, and so they must be added by regular addition on $\mathbb{R}$.

When simplifying this rule, we get,

$$u^{\lambda + \mu} = u^\lambda \cdot u^\mu,$$

which is a well-known exponential law.

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    $\begingroup$ Ohhh I see, yes I completely missed that and that makes sense. Thank you so much! $\endgroup$ – James Ronald Jan 11 '19 at 14:57

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