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Let G be a regular Lie group which is a directed union of regular Lie groups G i (all modelled on possibly infinite-dimensional, locally convex spaces). We show that G = lim

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We prove an implicit function theorem for Keller C k c -maps from arbitrary real or complex topological vector spaces to Fréchet spaces, imposing only a certain metric estimate on the partial differentials. As a tool, we show the C k -dependence of fixed points on parameters for suitable families of contractions of a Fréchet space. The investigations were stimulated by a recent metric approach to differentiability in Fréchet spaces by Olaf Müller. Our results also subsume generalizations of Müller's Inverse Function Theorem for mappings between Fréchet spaces. As an application, we study existence, uniqueness and parameter-dependence of solutions to suitable ordinary differential equations in Fréchet spaces.

We describe a setting of infinite-dimensional smooth (resp., analytic) Lie groups modelled on arbitrary, not necessarily sequentially complete, locally convex spaces, generalizing the framework of Lie theory formulated in [R. Hamilton, The inverse function theorem of Nash and Moser , Bull. Amer. Math. Soc. 7 (1982), 65-222] for Fréchet modelling spaces and in [J. Milnor, Remarks on infinite-dimensional Lie groups, in: B. DeWitt and R. Stora (eds.), Relativity, Groups and Topology II, North-Holland, 1983] for sequentially complete modelling spaces. Our studies were dictated by the needs of infinite-dimensional Lie theory in the context of the existence problem of universal complexifications. We explain why satisfactory results in this area can only be obtained if the requirement of sequential completeness is abandoned.