
TL;DR
This paper investigates the regularity properties of Milnor's infinite-dimensional Lie groups, establishing conditions for the continuity and differentiability of the evolution map in various topological settings, and exploring implications for Lie group integrability.
Contribution
It provides necessary and sufficient conditions for regularity of Milnor's Lie groups in $C^0$ and $C^k$ topologies, including new criteria for Mackey completeness and $ ext{k}$-confined curves.
Findings
The evolution map is $C^0$-continuous iff the group is locally $ extmu$-convex.
If the evolution map is defined on all smooth curves, then the group is Mackey complete.
Under local $ extmu$-convexity, $C^k$-curves are integrable iff the group is Mackey complete and $ ext{k}$-confined.
Abstract
We solve the regularity problem for Milnor's infinite dimensional Lie groups in the -topological context, and provide necessary and sufficient regularity conditions for the (standard) -topological setting. We prove that the evolution map is -continuous on its domain the Lie group is locally -convex. We furthermore show that if the evolution map is defined on all smooth curves, then is Mackey complete. Under the assumption that is locally -convex, we show that each -curve for is integrable (contained in the domain of the evolution map) is Mackey complete and -confined. The latter condition states that each -curve in the Lie algebra of can be uniformly approximated by a special type of sequence that…
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