The exponential law for spaces of test functions and diffeomorphism groups

We prove the exponential law A(E×F,G)≅A(E,A(F,G))A(E×F,G)≅A(E,A(F,G)) (bornological isomorphism) for the following classes AA of test functions: BB (globally bounded derivatives), W∞,pW∞,p (globally pp-integrable derivatives), SS (Schwartz space), DD (compact support), B[M]B[M] (globally Denjoy–Carleman), W[M],pW[M],p (Sobolev–Denjoy–Carleman), S[L][M] (Gelfand–Shilov), and D[M]D[M] (Denjoy–Carleman with compact support). Here E,F,GE,F,G are convenient vector spaces which are finite dimensional in the cases of DD, W∞,pW∞,p, D[M]D[M], and W[M],pW[M],p. Moreover, M=(Mk)M=(Mk) 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 DiffBDiffB, DiffW∞,pDiffW∞,p, DiffSDiffS, and DiffDDiffD are C∞C∞ Lie groups, and that DiffB{M}DiffB{M}, DiffW{M},pDiffW{M},p, DiffS{L}{M}, and DiffD{M}DiffD{M}, for non-quasianalytic MM, are C{M}C{M} Lie groups, where DiffA={Id+f:f∈A(Rn,Rn),infx∈Rndet(In+df(x))>0}DiffA={Id+f:f∈A(Rn,Rn),infx∈Rndet(In+df(x))>0}. We also discuss stability under composition.