Inductive limit of general linear groups (Q1296416)

The main results of the paper are: (i) For the topology induced from the inductive locally convex vector topology of \(M(\Lambda)\), \(GL(\Lambda)\) is isomorphic to \(C(X,GL(C))\) as topological groups. (ii) \(G\) becomes a topological group with the inductive limit topology as topological spaces if \(\forall n\), \(\exists U\), \(\exists m>n\) such that \(\overline U^{(m)}\) is compact, where \(U\) is a neighbourhood of 1 and \(\overline U^{(m)}\) is the closure of \(U\) in \(G_m\). (iii) If the injection \(G_n\hookrightarrow G_{n+1}\) is a homeomorphism and \(G_1\) is open in \(G_n\) for all \(n\), then the inductive limit topology is a group topology.