# Why is the set of all positive real numbers with addition defined by $x + y = xy$ and scalar multiplication defined by $cx = x^c$ a vector space?

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.

• check cu + du you forgot a step – T. Fo Jan 11 '19 at 14:13
• 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. – lulu Jan 11 '19 at 14:15
• But $cu$ and $du$ are vectors and vectors sum is $x * y$. This means that $cu+du=2^3 * 2^4=2^7$. – 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.

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