Defining operations for vector-space Let $R^+=\{x \in R:x>0\}$. Consider $V=R^+$ and the field $F=R$. Define the addition and scalar multiplication in a way that $V$ it's a vector-space over $F$.
My answer: Let x and y be elements of V. We have that $x+y \in R^+=V$ because addition is closed for the positive real numbers.
Let $a \in R$ and $x \in V$
We have three cases for the product ax:
Case 1) $a>0$ that means $a \in R^+$
Hence $ax\in R^+$ by closedure.
Case 2) $a=0$
$ax=0 \in R$
Case 3)$a<0$
Therefore $ax \notin R^+$
Then V is a vector-space over R.
Is my answer correct or am I missing something?

You have been asked to define an operation $\oplus:\Bbb R^+\times\Bbb R^+\to\Bbb R^+$ and an operation $\odot :\Bbb R\times \Bbb R^+\to\Bbb R^+$ such that $(\Bbb R^+,\oplus,\odot)$ is a $\Bbb R$-vector space. From a sheer semantical standpoint, you have not done that because nowhere in your text you say: "I take $\oplus$ to be this operation and $\odot$ to be this operation".
As you have said, the two operations you have mentioned cannot, in any order, appear in the triple $(\Bbb R^+,\oplus,\odot)$, because one of the two does not have range in $\Bbb R^+$.
