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# Completeness of infinite-dimensional Lie groups in their left uniformity

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• Helge Glöckner,
Institut für Mathematik, Universität Paderborn , Warburger Str. 100, D-33098 Paderborn, Germany
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## Abstract

We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_1\subseteq G_2\subseteq\cdots$ of topological groups~$G_n$ such that~$G_n$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on~$G_n$, for each $n\in\mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each~$G_n$ whenever $\bigcup_{n\in \mathbb{N}}V_1V_2\cdots V_n$ is an identity neighbourhood in~$G$ for all identity neighbourhoods $V_n\subseteq G_n$. If, moreover, each $G_n$ is complete, then~$G$ is complete. We also show that the weak direct product $\bigoplus_{j\in J}G_j$ is complete for each family $(G_j)_{j\in J}$ of complete Lie groups~$G_j$. As a consequence, every strict direct limit $G=\bigcup_{n\in \mathbb{N}}G_n$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\operatorname{Diff}_c(M)$ of a paracompact finite-dimensional smooth manifold~$M$ and the test function group $C^k_c(M,H)$, for each $k\in\mathbb{N}_0\cup\{\infty\}$ and complete Lie group~$H$ modelled on a complete locally convex space.
 Keywords: infinite-dimensional Lie group, left uniform structure, completeness
 MSC Classifications: 22E65 - Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05] 22A05 - Structure of general topological groups 22E67 - Loop groups and related constructions, group-theoretic treatment [See also 58D05] 46A13 - Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40] 46M40 - Inductive and projective limits [See also 46A13] 58D05 - Groups of diffeomorphisms and homeomorphisms as manifolds [See also 22E65, 57S05]

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