I studied differential geometry last semester and since I am quite into functional analysis I questioned myself if one could expand the theory to Hilbert or Banach space settings since you could use the Frechet derivative to make similiar definitions. So I just used google and found a Wikipedia article on Banach manifolds. They are the kind of objects I had in mind but there were given some examples which seemed to be rather trivial.
Examples given were: Banach spaces are Banach manifold, open subsets of Banach spaces are Banach manifolds. Both these examples facilitate the identity map as chart and are thus pretty trivial.
So my question is if there are non-trivial examples that aren't pathological in the sense that they are not just interesting as non-trivial examples. Are there real applications for that theory or is it just some kind of abstract nonsense. I would really appreciate some input :)
For Hilbert manifold, loops spaces of usual manifolds are important examples. The space of $H^{1,2}$ loops, for instance ( loops with one derivative in $L^2$). Another example is the case of loops spaces of a given submanifold of $\bf R^n$. Continuous loops in $\bf R^n$ is a $L^2$ (Hilbert) space (Fourier series). The set of loops contained in a submanifold is a sub-Hilbert manifold.