The exponential law for spaces of test functions and diffeomorphism groups

Abstract

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