Let $f_1:\mathbb R\to \mathbb R$ be a Schwartz function, and $\mathcal S(\mathbb R)$ be the space of Schwartz functions (Schwartz space). $\mathcal B(\mathcal S)$ is the Borel $\sigma$-field of $\mathcal S$. Let $f_2: \mathcal S(\mathbb R)\to\mathbb R$ be a function-valued function, something like (not formal since if I know how to formally write this, I won't be asking this question):
$$\sup_{f_1\in\mathcal S_1(\mathbb R)}(1+|f_1|)^m|Df_2|<\infty \ \ \forall m\in\mathbb N$$
Let's call this a "Schwartz function" of Schwartz function, and denote the space as $\mathcal S_2(\mathbb R)=\mathcal S(\mathcal S(\mathbb R))$.
The questions are
Are there any works done related to this?
How to rigorously define a topology and measure over $\mathcal S_n(\mathbb R)$ ?
Some mathematicians said that it is related to Hida distributions, Kondratiev triple, Borchers algebra, and quantum field theory.