2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70361We show that every countable direct system of finite-dimensional real or complex Lie groups has a direct limit in the category of Lie groups modelled on locally convex spaces. This enables us to push all basic constructions of finite-dimensional Lie theory to the case of direct limit groups. In particular, we obtain an analogue of Lie's third theorem: Every countable-dimensional real or complex locally finite Lie algebra is enlargible, i.e., it is the Lie algebra of some regular Lie group (a suitable direct limit group).33 pages (v2: Lemma 7.12 and Proposition 7.13 corrected, clearer distinction between analyticity and convenient analyticity)Group Theory22E65 (Primary) 46T05, 57N40, 58B10, 58B25 (Secondary)text