Let G be a regular Lie group which is a directed union of regular Lie groups G_i (all modelled on possibly infinite-dimensional, locally convex spaces). We show that G is the direct limit of the G_i as a regular Lie group whenever G admits a so-called direct limit chart. Notably, this allows the regular Lie group Diff_c(M) of compactly supported smooth diffeomorphisms to be interpreted as a direct limit of the regular Lie groups Diff_K(M) of smooth diffeomorphisms supported in compact subsets K of M, even if the finite-dimensional smooth manifold M is merely paracompact (but not necessarily sigma-compact), which was not known before. Similar results are obtained for the test function groups C^k_c(M,F) with values in a Lie group F.

中文翻译：

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正则李群的直接极限

令 G 是一个正则李群，它是正则李群 G_i 的有向并集（都建模在可能的无限维局部凸空间上）。我们证明 G 是 G_i 作为正则李群的直接极限，只要 G 承认所谓的直接极限图。值得注意的是，这允许紧支持的光滑微分同胚的正则李群 Diff_c(M) 被解释为 M 的紧子集 K 中支持的光滑微分同胚的正则李群 Diff_K(M) 的直接极限，即使有限维光滑流形 M 只是超紧致（但不一定是 sigma-紧致），这在以前是未知的。对于李群 F 中的值的测试函数群 C^k_c(M,F) 获得了类似的结果。