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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 :)

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    Wikipedia tells me the following: A 1969 theorem of David Henderson states that every infinite-dimensional, separable, metric Banach manifold X can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, H (up to linear isomorphism, there is only one such space). In fact, Henderson's result is stronger: the same conclusion holds for any metric manifold modeled on a separable infinite-dimensional Fréchet space. https://en.wikipedia.org/wiki/Banach_manifold#Classification_up_to_homeomorphism https://en.wikipedia.org/wiki/Hilbert_manifold – 2017-02-19
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    ...so really the "only" Banach manifolds are open subsets of Hilbert spaces. – 2017-02-19
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    Ok, I must have overread that. Thank you :) – 2017-02-19

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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.