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We develop aspects of geometric control theory on Lie groups G which may be infinite dimensional, and on smooth G -manifolds M modelled on locally convex spaces. As a tool, we discuss existence and uniqueness questions for differential equations on M given by time-dependent fundamental vector fields which are L 1 in time. We then discuss the closures of reachable sets in M for controls in the Lie algebra g of G , or within a compact convex subset of g. Regularity properties of the Lie group G play an important role. [ABSTRACT FROM AUTHOR]

We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter k R -spaces and locally convex spaces E such that E × E is a k R -space. [ABSTRACT FROM AUTHOR]

Recall that a topological space X is said to be a k ω -space if it is the direct limit of an ascending sequence K 1 ⊆ K 2 ⊆ ⋯ of compact Hausdorff topological spaces. If each point in a Hausdorff space X has an open neighbourhood which is a k ω -space, then X is called locally k ω . We show that a topological group is complete whenever the underlying topological space is locally k ω . As a consequence, every infinite-dimensional Lie group modelled on a Silva space is complete. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a Lie group which is the union of an ascending sequence of Lie groups (all of which may be infinite-dimensional). We study the question when in the category of Lie groups, topological groups, smooth manifolds, respectively, topological spaces. Full answers are obtained for G the group of compactly supported -diffeomorphisms of a σ-compact smooth manifold M; and for test function groups of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups. [Copyright &y& Elsevier]

Abstract: We show that countable direct limits of finite-dimensional Lie groups do not have small subgroups. The same conclusion is obtained for suitable direct limits of infinite-dimensional Lie groups. [Copyright &y& Elsevier]

Abstract: Let be a homomorphism between smooth Lie groups modelled on Mackey complete, locally convex real topological vector spaces. We show that if f is Hölder continuous at 1, then f is smooth. [Copyright &y& Elsevier]