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Completeness of infinitedimensional Lie groups in their left uniformity


Helge Glöckner,
Institut für Mathematik, Universität Paderborn , Warburger Str. 100, D33098 Paderborn, Germany
Abstract
We prove completeness for the main examples
of infinitedimensional 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 finitedimensional
Lie groups is complete, as well as the diffeomorphism group
$\operatorname{Diff}_c(M)$
of a paracompact finitedimensional 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.
MSC Classifications: 
22E65, 22A05, 22E67, 46A13, 46M40, 58D05 show english descriptions
Infinitedimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05] Structure of general topological groups Loop groups and related constructions, grouptheoretic treatment [See also 58D05] Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40] Inductive and projective limits [See also 46A13] Groups of diffeomorphisms and homeomorphisms as manifolds [See also 22E65, 57S05]
22E65  Infinitedimensional 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, grouptheoretic 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]
