Differentiability along one-parameter subgroups compared to differentiability on Lie groups as manifolds
Volume 113 / 2017
Banach Center Publications 113 (2017), 363-373
MSC: Primary 22E65; Secondary 22A10, 26E15, 46E50.
DOI: 10.4064/bc113-0-17
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), Beltiţă and Nicolae (2015), and others, in varying degrees of generality. A notion of $C^{r,s}$-functions on products $G\times H$ of topological groups was introduced by Nikitin (2016). We recall this concept and an exponential law of the form $C^{r,s}(G\times H,E)\cong C^r(G,C^s(H,E))$ (under suitable hypotheses on $G$ and $H$). Furthermore, we show that in the case where $G$ is a locally exponential Lie group or a certain direct limit Lie group our calculus of $C^r$-functions coincides with the differential calculus on $G$ as a locally convex manifold.
