Exponential laws for spaces of differentiable functions on topological groups

TL;DR

This paper investigates the structure of spaces of smooth functions on topological groups, establishing exponential laws that relate function spaces on product groups to iterated function spaces, under certain conditions.

Contribution

It proves isomorphisms between spaces of smooth functions on product groups and iterated function spaces, extending previous results to broader classes of topological groups.

Findings

Isomorphism between $C^ abla(G\times H,E)$ and $C^ abla(G,C^ abla(H,E))$ for metrizable or locally compact groups.

Identification of $C^k(G, C^l(H,E))$ with functions on $G\times H$.

Extension of exponential laws to spaces of differentiable functions on topological groups.

Abstract

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 $G\times H$.