Let $\mathcal{S}(\mathbb{R}^{n})$ be the Schwartz space of all smooth functions $f \in C^{\infty}(\mathbb{R}^{n})$ such that $||f||_{\alpha, \beta} \le \infty$, $$||f||_{\alpha, \beta} = \sup_{x \in \mathbb{R}^{n}}{|x^{\alpha}D^{\beta}(f(x))|}, \forall \alpha, \beta \in \mathbb{Z}_{+}$$ The topology is induced by the familiy of the seminorms above, transforming this space into a locally convex topological vector space.
I was told that it is possible to construct a topological isomorphism (i.e. a continuous homomorphism with continuous inverse) between the Schwartz space $\mathcal{S}({\mathbb{R}^{n}})$ and $$D(a) = \{ f \in C^{\infty}(S^{n}): Df(p) = 0, \forall D \in A \}$$ where $A$ stands for the algebra of linear operators, generated by the vector fields (i.e. $A$ is the algebra of derivations, $D: C^{\infty}({S^{n}}) \rightarrow C^{\infty}({S^{n}})$ such that $D(fg) = D(f)g+fD(g)$ for $f, g \in C^{\infty}(S^{n})$) and $p \in S^{n}$ is some fixed point on a sphere.
I am familiar with some concrete, but still general examples, in which, for example, the space of sections of a vector bundle is endowed with Sobolev norms, whilst the latter naturally arises in the branch of differential geometry that studies the interconnection between analytical and topological indices (Atyah-Singer theorem), but here i have bumped into a sort of confusion.
Are there any ways possible to treat the problem described just above?
(probably, this is one of the easiest way to show that the Schwartz space is nucreal, though personally i'm more interested in the original problem as formulated).