The exponential law for spaces of test functions and diffeomorphism groups

We prove the exponential law A(E x F, G) congruent to A(E, A(F, G)) (bornological isomorphism) for the following classes A of test functions: B (globally bounded derivatives), W-infinity,W-P (globally p-integrable derivatives), S (Schwartz space), 1) (compact support), B-[M] (globally Denjoy-Carleman), W-[M],W-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-infinity,W-P, D-[M], and W-[M],W-P. Moreover, 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 Diff B, Diff W-infinity,W-P, Diff S, and Diff D are C-infinity Lie groups, and that Diff B-{M}, Diff W-{M},W-P, Diff S-{L}({M}), and Diff D-{M}, for non-quasianalytic M, are C-{M} Lie groups, where Diff A = {Id +f : f is an element of A(R-n , R-n), inf(x is an element of Rn) det (IIn + df (x)) > 0}. We also discuss stability under composition. (C) 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.