Background: Related question
I am trying to prove, that the countably-normed spaces $C^\infty_0(B_1)$ on the open unit ball (i.e. function and all derivatives vanish at the boundary) in $\mathbb{R}^n$ and the Schwartz Space $\mathscr{S}(\mathbb{R}^n)$ are isomorphic. Here, $\mathscr{S}$ carries the usual Schwartz topology and $C^\infty_0(B_1)$ the locally convex topology induced by the family $\|\varphi\|_i=\sup_{x\in B_1}\max_{|\alpha|\leq i}|D^\alpha\varphi|$.
My ansatz is to use, that the open unit ball and whole $\mathbb{R}^n$ are diffeomorphic, e.g. via the diffeomorphism $$ h:x\rightarrow \frac{x}{\sqrt{1+|x|^2}} $$ therefore my guess for the isomorphism is $$ I:C^\infty_0(B_1)\rightarrow \mathscr{S}(\mathbb{R}^n), f\rightarrow f\circ h $$ However, I am unable to prove, that $I$ and $I^{-1}$ are continuous with respect to the locally convex topology.
Concretely, I have to show that for every pair $p,q\in \mathbb{N}$ there exist $r\in \mathbb{N},C>0$ with $$ \sup_{x\in \mathbb{R}^n}\max_{\alpha\leq q} (1+|x|^2)^p|D^\alpha (\varphi \circ h) | \leq C \sup_{x\in B_1}\max_{\alpha\leq r} |D^\alpha \varphi | $$ for all $\varphi\in C_0^\infty(B_1)$, similar for $I^{-1}$. The crux about the continuity has to be, that the vanishing at the boundary of $B_1$ for some function $\varphi$ somehow also implies the vanishing at infinity for $I(\varphi)$, however, I am not able to formulate a clean proof for this.
Does anybody have an idea or a reference? I found a similar statement in Treves (Topological Vector Spaces) Theorem 51.4, unfortunately, the proof of the theorem was "left as exercise", and I am obviously too dumb to solve it.
I would greatly appreciate any help!