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Is their a generalisation of the Hopf–Rinow theorem to infinite dimensional manifolds (Frechet, Banach or Hilbert)?

I have actually never seen the prove of the Hopf–Rinow theorem so i don't know if their are obvious generalisations.

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    $\begingroup$ The direct generalization is false - you can find a discussion of the failure in Chapter VIII of Lang's Fundamentals of Differential Geometry, along with a different sufficient condition for completeness in the case of Hilbert manifolds. If you're looking for generalizations to particular settings then it should be fairly googleable. $\endgroup$ – Anthony Carapetis Mar 12 '18 at 10:04
  • $\begingroup$ @AnthonyCarapetis Thank you for the reference $\endgroup$ – Netivolu Mar 12 '18 at 10:30

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