Can you recommend me of books or preprints that cover reasonabely well this topic?
Do you mean the basic study of Banach manifolds h(ttp://en.wikipedia.org/wiki/Banach_manifold)?
In any case, it cannot hurt for you to take a look at the book (free online) "The convenient setting for global analysis" which is a book about infinite dimensional manifolds.
Basically I am looking for DG on a more general setting than the one covered in any first course in DG, I guess Hilbert spaces and banach spaces comes next after the real space.
I'll check the book.
Well, I now find myself in similar circumstances. I want a textbook on the most general geometric objects possible, but I don't know how much more general you can get then Banach manifolds. I guess the algebraic stuff?
a similar question came up on mathoverflow.
Lang's "Fundamentals of Differential Geometry"
And Abraham/Marsden/Ratiu's "Manifolds, Tensor Analysis and applications"
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