Does the infinite-dimensional separable Hilbert space over $\Bbb C$ form a Lie group? It is a Banach space, that is, a complete normed space.
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2026-05-05 21:35:00.1778016900
Is the separable infinite-dimensional Hilbert space over $\Bbb C$ a Lie group?
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It depends on the definitions you take. The Hilbert space $H$, with the addition operation and the topology induced by the inner product, is a topological group. To be a Lie group, it should also be a manifold. If you take "manifold" to mean "locally homeomorphic to a linear space" then of course $H$ satisfies this (globally!). But the usual defininitions of manifold require the charts to map into finite-dimensional linear space; this would require $H$ to be locally compac, which it isn't.