Vector Space Structures over ($\mathbb{R}$,+)

Question

Consider the abelian group ($\mathbb{R}$,+) of real numbers with the usual addition. Is there a scalar multiplication \begin{equation} \cdot : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}, \end{equation} other than the usual multiplication, which makes ($\mathbb{R}$,+,$ \cdot $) a real vector space?

Answer 1

Let us denote by $m \colon \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ the multiplication we are looking for and assume that $(\mathbb{R}, +, m)$ is a real vector space. Assuming the axiom of choice, every real vector space has a basis and so there must be an isomorphism of real vector spaces $\varphi \colon (\bigoplus_{i \in I} \mathbb{R}, +, \cdot) \rightarrow (\mathbb{R}, +, m)$. Then

$$ \varphi(\alpha \cdot v) = m(\alpha, \varphi(v)) $$

and so the scalar multiplication is given by

$$m(\alpha, x) = \varphi(\alpha \cdot \varphi^{-1}(x)). $$

Now, we can try and reverse this process. Let $\varphi \colon (\mathbb{R}^2, +) \rightarrow (\mathbb{R}, +)$ be an isomorphism of abelian groups (see this answer) and define a scalar multiplication on $\mathbb{R}$ by the formula

$$m(\alpha, x) = \varphi(\alpha \cdot \varphi^{-1}(x)) $$

where $\cdot$ is the standard scalar multiplication $\cdot \colon \mathbb{R} \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Intuitively, we are transferring the standard scalar multiplication on $(\mathbb{R}^2, +, \cdot)$ to $(\mathbb{R}, +)$ using $\varphi$. You can easily check that $m$ satisfies the relevant four axioms to be a scalar product. Finally, if $m(\alpha, x) = \alpha x$ then

$$ \alpha x = \varphi(\alpha \cdot \varphi^{-1}(x)) \implies \varphi^{-1}(\alpha x)= \alpha \cdot \varphi^{-1}(x)$$

so $m$ is the standard multiplication if and only if $\varphi^{-1}$ is not only an additive isomorphism but also an isomorphism of vector spaces. This is not possible for dimension reasons so this indeed defines a different vector space structure on $(\mathbb{R}, +)$ and all such structures come from this construction. You can also take an additive non-linear isomorphism $\varphi \colon (\mathbb{R}, +) \rightarrow (\mathbb{R}, +)$ and apply the construction using this $\varphi$. In any case, after performing the construction the additive isomorphism $\varphi$ becomes by definition an isomorphism of real vector spaces.