Exponential laws for spaces of differentiable functions on topological groups

Smooth functions $f:G\to E$ from a topological group $G$ to a locally convex space $E$ were considered by Riss (1953), Boseck, Czichowski and Rudolph (1981), Belti\c{t}\u{a} and Nicolae (2015), and others, in varying degrees of generality. The space $C^\infty(G,E)$ of such functions carries a natural topology, the compact-open $C^\infty$-topology. For topological groups $G$ and $H$, we show that $C^\infty(G\times H,E)\cong C^\infty(G,C^\infty(H,E))$ as a locally convex space, whenever both $G$ and $H$ are metrizable or both $G$ and $H$ are locally compact. Likewise, $C^k(G, C^l(H,E))$ can be identified with a suitable space of functions on $G\times H$.