We solve the regularity problem for Milnor’s infinite dimensional Lie groups in the C~0-topological context, and provide necessary and sufficient regularity conditions for the (standard) Ck- topological setting. Specifically, we prove that if G is an infinite dimensional Lie group in Milnor’s sense, then the evolution map is C~0-continuous on its domain iff G is locally μ-convex – This is a continuity condition imposed on the Lie group multiplication that generalizes the triangle inequality for locally convex vector spaces. We furthermore show that if the evolution map is defined on all smooth curves, then G is Mackey complete – This is a completeness condition formulated in terms of the Lie group operations that generalizes Mackey completeness as defined for locally convex vector spaces; so that we generalize the well known fact that a locally convex vector space is Mackey complete if each smooth (compactly supported) curve is Riemann integrable.
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