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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$H^{>m/2}(M,G)$ be the group of all mappings $\gamma\colon M\to G$ which are$H^s$ for some $s>m/2$. We show that $H^{>m/2}(M,G)$ can be made a regular Liegroup in Milnor's sense, modelled on the Silva space $H^{>m/2}(M,g)$ which isthe locally convex direct limit of the Hilbert spaces $H^s(M,g)$ for $s>m/2$,such that $H^{>m/2}(M,G)$ is the direct limit of the Hilbert-Lie groups$H^s(M,G)$ for $s>m/2$ 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 ageneral construction of Lie groups $F(M,G)$ whenever real-valued functionspaces $F(U,R)$ on open subsets $U$ of $R^m$ are given, subject to simpleaxioms.

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