Mapping groups associated with real-valued function spaces and direct limits of Sobolev-Lie groups

Let $M$ be a compact smooth manifold of dimension $m$ (without boundary) and $G$ be a finite-dimensional Lie group, with Lie algebra $g$. Let $H^{>m\mathrm{/}2}(M,G)$ be the group of all mappings $\gamma \text{\hspace{0.17em}⁣}:M\to G$ which are $H^s$ for some $s>m\mathrm{/}2$. We show that $H^{>m\mathrm{/}2}(M,G)$ can be made a regular Lie group in Milnor’s sense, modelled on the Silva space $H^{>m\mathrm{/}2}(M,g)$ which is the locally convex direct limit of the Hilbert spaces $H^s(M,g)$ for $s>m\mathrm{/}2$, such that $H^{>m\mathrm{/}2}(M,G)$ is the direct limit of the Hilbert-Lie groups $H^s(M,G)$ for $s>m\mathrm{/}2$ as a smooth Lie group. We also explain how the (known) Lie group structure on $H^s(M,G)$ can be obtained as a special case of a general construction of Lie groups $F(M,G)$ whenever real-valued function spaces $F(U,R)$ on open subsets $U$ of $R^m$ are given, subject to simple axioms.