Hilbert manifold - definition *

Even if one is interested only in finite-dimensional manifolds, the need for infinitedimensional manifolds sometimes arises. For example, one approach to study closed geodesics on a manifold is to use Morse theory on its (free) loop space; while for some purposes it is enough to work with finite-dimensional approximations, it is helpful for some finer aspects of the theory to use models of the free loop space that are infinite-dimensional manifolds. The use of Morse theory in an infinite-dimensional context is even more important for other (partial) differential equations like those occuring in the theory of minimal surfaces and the Yang-Mills equations. Morse theory for infinite dimensional manifolds was developed by Palais and Smale ([20], [21]). While there is up to isomorphism only one vector space of every finite dimension, there are many different kinds of infinite-dimensional topological vector spaces one can choose. Modeling spaces on Fréchet spaces gives the notion of Fréchet manifolds, modelling on Banach spaces gives Banach manifolds, modelling on the Hilbert cube (the countably infinite product of intervals) gives Hilbert cube manifolds. We will stick to Hilbert manifolds (which are not directly related to Hilbert cube manifolds).