1-Torus as finite dimensional $\mathbb{R}$-vector space is one dimensional, yet not isomorphic to $\mathbb{R}$

In fact, the $1$-torus fails to be a vector space over $\Bbb R$. Following the list of axioms given here, the torus fails "compatibility of scalar multiplication with field multiplication". Note for instance that $$ \frac 14 \odot (4 \odot R_{\pi/2}) = R_0 \neq (\frac 14 \cdot 4) \odot R_{\pi/2}. $$

The putative scalar multiplication map, $$r \odot R_{\theta} \mapsto R_{r \theta} ,$$ is not even well-defined.

The periodicity of $\sin, \cos$ impliy that $$R_{\theta + 2\pi} = R_\theta .$$ But taking (for notational convenience) $\theta = 2 \beta$ and symbolically applying the rule for the scalar multiplication map (i.e., temporarily not worrying about well-definedness) gives that $$\frac{1}{2} \odot R_{2 \beta + 2\pi} = R_{\beta + \pi} = R_\beta R_\pi = - R_\beta,$$ which does not coincide with $$\frac{1}{2} \cdot R_{2 \beta} = R_{\beta} .$$

Putting this a little more abstractly (and formally): The map $\pi : \theta \mapsto R_{\theta}$ is a quotient map and identifies $\Bbb T$ with the space $\Bbb R / \sim$, where $x \sim y$ iff $\pi(x) \leftrightarrow \pi(y)$.

• The addition operation $+$ of the real vector space $\Bbb R$ descends via $\pi$ to an operation on $\Bbb T$, namely, $\oplus$. It follows that $\oplus$ satisfies the usual axioms of the vector space addition, and in particular $(\Bbb T, \oplus)$ is a group (isomorphic to $SO(2)$). (In fact, $\pi$ is a group homomorphism $(\Bbb R, +) \to (\Bbb T, \oplus)$.)
• On the other hand, the scalar multiplication operation $\cdot : \Bbb R \times \Bbb R \to \Bbb R$ does not descend to a map $\Bbb R \times \Bbb T \to \Bbb T$: As the above computation shows, $\pi(r \cdot \alpha)$ is not independent of the choice of representative $\alpha$ of $R_\theta$ in $\pi^{-1}(R_\theta) = \{\theta + 2 \pi k : k \in \Bbb Z\}$. But this descent was how the map $\odot$ was characterized, so it is not well-defined.