Let $\mathbb{R}^n$ be $n$-dimensional Euclidean space and $S^n$ be $n$-sphere.
Then, it is well-known that $S^n$ is the one-point compactification of $\mathbb{R}^n$.
Now consider $C^\infty(S^n)$, the Frechet space of smooth functions on $S^n$.
Then, my question is
Which function space on $\mathbb{R}^n$ is identified with $C^\infty(S^n)$?
According to this link, the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is identified with a subspace of $C^\infty(S^n)$ comprising functions whose derivatives all vanish at a fixed point of $S^n$.
So, my current question is sort of inverse to the above link. I suspect that the answer to be
Smooth functions on $\mathbb{R}^n$ with bounded derivatives.
However, I am not sure if this is right. Could anyone help me?