The exponential law for spaces of test functions and diffeomorphism groups

We prove the exponential law $\mathcal A(E \times F, G) \cong \mathcal A(E,\mathcal A(F,G))$(bornologicalisomorphism)forthefollowingclasses$\mathcal A$oftestfunctions:$\mathcal B$(globallyboundedderivatives),$W^{\infty,p}$(globally$p$-integrablederivatives),$\mathcal S$ (Schwartz space), $\mathcal{D}$ (compact sport, ${\mathcal{B}}^{[M]}$ (globally Denjoy_Carleman), $W^{[M],p}$ (Sobolev_Denjoy_Carleman), $\mathcal S_{[L]}^{[M]}$(Gelfan{d}_{S}hilov),and$\mathcal D^{[M]}$\mathrm{.}Here$E, F, G$ are convenient vector spaces (finite dimensional in the cases of $W^{\mathrm{\infty },p}$, $\mathcal{D}$, $W^{[M],p}$, 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 $\mathrm{Diff}\mathcal{B}$, $\mathrm{Diff}{W}^{\mathrm{\infty },p}$, $\mathrm{Diff}\mathcal{S}$, and $\mathrm{Diff}\mathcal{D}$ are $C^{\mathrm{\infty }}$ Lie groups, and $\mathrm{Diff}{\mathcal{B}}^{\{M\}}$, $\mathrm{Diff}{W}^{\{M\},p}$, $\operatorname{Diff} \mathcal S_{{L}}^{{M}}$,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.