Given an arbitrary Hilbert manifold, can on find a complete Riemannian metric and a Morse function satisfying the PalaisSmale condition?
I recently learned that the answer to the question is YES, answered in the ETH preprint "Hcobordism for Hilbert Manifolds" by Dan Burghelea. I found the reference in the article "On the differential topology of Hilbert manifolds" of Eells and Elworthy. 


Well, I have been away from this kind of question for a long while, so please don't the following remarks as definitive in any way, but I am not aware of any counterexample and in the infinite dimensional case I also do not know any positive result. So, I think it is a nice question you are asking and if you can prove something in this direction it would be interesting and probably publishable. A relatively minor pointI assume that you mean a differentiable Hilbert manifold, and you probably want to assume the manifold is separable. With those assumption I believe it is not hard to show that the manifold can be smoothly embedded as a closed submanifold of Hilbert space, so it gets a complete Riemannian metric, and it would be really nice if you could show that for some such embedding there was a ``height function'' (i.e., a continuous linear functional on the Hilbert space) that when restricted to the manifold satisfied Condition C. If I had to bet, I would guess this is so. 

