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Why are Banach and Frechet manifolds studied not even remotely as much as Euclidean manifolds? I assume like many other mathematical subjects, theory of manifolds has been developed much more than the real world needs. So perhaps applicability is not a factor and there are some mathematically intrinsic reasons for unpopularity of Banach manifolds comparing to ordinary manifolds.

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They are popular (in the sense of: people would like to use them, some methods of dealing with nonlinear PDE are based on the theory, for example), but in some respects more difficult to handle than the finite dimensional manifolds. B-Space can be, e.g., non-reflexive, which requires additional care when you want to study the cotangent space. Also, in infinite dimensions, compactness (which is one highly important ingredient in existence proofs) is a rather complex issue. This means, in particular, that they are often only treated/used in advanced courses.

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Good to know. How about Hilbert manifold? –  Troy Woo Sep 2 at 5:35
    
Hilbert spaces are reflexive, but the compactness issues are still there. –  Thomas Sep 2 at 6:06
    
Serge Lang write a book on them trying to do as much as possible of the standard differential topology in the Banach/Hilbert setting. –  studiosus Sep 4 at 20:32

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