Direct limits of regular Lie groups

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 = lim ⟶G ias a regular Lie group if  G admits a so‐called direct limit chart. Notably, this allows the regular Lie groupDiff c ( M )of compactly supported diffeomorphisms to be interpreted as a direct limit of the regular Lie groupsDiff K ( M )of diffeomorphisms supported in compact sets K ⊆ M , even if the finite‐dimensional smooth manifold  M is merely paracompact (but not necessarily σ‐compact), which is new. Similar results are obtained for the test function groupsC c k ( M , F )with values in a Lie group  F .