The exponential law for spaces of test functions and diffeomorphism groups

We prove the exponential law $mathcal A(E times F, G) cong mathcal A(E,mathcal A(F,G))$ (bornological isomorphism) for the following classes $mathcal A$ of test functions: $mathcal B$ (globally bounded derivatives), $W^{infty,p}$ (globally $p$-integrable derivatives), $mathcal S$ (Schwartz space), $mathcal D$ (compact sport, $mathcal B^{[M]}$ (globally Denjoy_Carleman), $W^{[M],p}$ (Sobolev_Denjoy_Carleman), $mathcal S_{[L]}^{[M]}$ (Gelfand_Shilov), and $mathcal D^{[M]}$. Here $E, F, G$ are convenient vector spaces (finite dimensional in the cases of $W^{infty,p}$, $mathcal D$, $W^{[M],p}$, and $mathcal D^{[M]})$, and $M=(M_k)$ is a weakly log-convex weight sequence of moderate growth. As application we give a new simple proof of the fact that the groups of diffeomorphisms $operatorname{Diff} mathcal B$, $operatorname{Diff} W^{infty,p}$, $operatorname{Diff} mathcal S$, and $operatorname{Diff}mathcal D$ are $C^infty$ Lie groups, and $operatorname{Diff} mathcal B^{{M}}$, $operatorname{Diff}W^{{M},p}$, $operatorname{Diff} mathcal S_{{L}}^{{M}}$, and $operatorname{Diff}mathcal D^{[M]}$, for non-quasianalytic $M$, are $C^{{M}}$ Lie groups, where $operatorname{Diff}mathcal A = {operatorname{Id} +f : f in mathcal A(mathbb R^n,mathbb R^n), inf_{x in mathbb R^n} det(mathbb I_n+ df(x))>0}$. We also discuss stability under composition.