Completeness of Infinite-dimensional Lie Groups in Their Left Uniformity

We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence G(1)subset of G(2) subset of ... 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 is an element of 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 boolean OR(n)(is an element of N) V1V2 ... V-n is an identity neighbourhood in G for all identity neighbourhoods V-n subset of G(n). If, moreover, each G(n) is complete, then G is complete. We also show that the weak direct product circle plus(j)(is an element of J) G(j) is complete for each family (G(j))(j)(is an element of J) of complete Lie groups G(j). As a consequence, every strict direct limit G = boolean OR(n)(is an element of N) G(n) of finite-dimensional Lie groups is complete, as well as the diffeomorphism group Diff(c) (M) of a paracompact finite-dimensional smooth manifold M and the test function group C-c(k) (M, H), for each k is an element of N-0 boolean OR {infinity} and complete Lie group H modelled on a complete locally convex space.