# existence of Morse functions satisfying the Palais-Smale condition

Given an arbitrary Hilbert manifold, can on find a complete Riemannian metric and a Morse function satisfying the Palais-Smale condition?

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A relatively minor point---I 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.