Basically what the title says — if a Lie group is infinite-dimensional, is it necessarily noncompact?
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12$\begingroup$ I'm not sure the notion of infinite-dimensional Lie group is entirely standard, so you might want to provide the particular definition you are working with (or at least cite a reference with the precise definition). $\endgroup$– Sam HopkinsCommented Nov 20, 2023 at 22:45
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$\begingroup$ @SamHopkins I suppose roughly the structure I am imagining is an infinite dimensional smooth manifold (in the sense of being locally homeomorphic to a Banach space) and endowed with a group structure. I am coming from physics rather than math so I imagine roughly that the most restrictive possible structure consistent with a nonempty set of interesting examples is appropriate, but I admit I don't have a perfectly precise notion of what should be in this category. $\endgroup$– PanopticonCommented Nov 20, 2023 at 23:16
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14$\begingroup$ It's not locally compact, hence not compact. $\endgroup$– YCorCommented Nov 20, 2023 at 23:21
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4$\begingroup$ What @YCor said. An infinite-dimensional Banach space cannot be locally compact. $\endgroup$– Gerald EdgarCommented Nov 21, 2023 at 2:13
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