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Hi,

I am looking for a book on Banach manifolds. Can somebody recommend me something. Thanks in advance.

leo

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For Banach-Lie groups, I recommend the Springer Lecture Notes 285 by P. de la Harpe: Classical Banach-Lie algebras and Banach-Lie groups of operators in Hilbert space / Pierre de la Harpe / Berlin : Springer-Verlag , 1972 –  Alain Valette Nov 13 '11 at 15:57
    
You could find helpful the answers received by this previous similar question: mathoverflow.net/questions/74298/… –  Giuseppe Tortorella Nov 13 '11 at 17:45
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It might help if you gave a little background and motivation. There are many books that treat Banach manifolds, but at different levels. If just starting out, for example, then several standard differential geometry texts treat Banach manifolds from the outset (I think that Lang's book is one of these). –  Andrew Stacey Nov 14 '11 at 8:19
    
For the possible benefit of other people interested in Banach manifolds, perhaps the banach-manifolds tag can be added to this question? –  Philip Brooker Nov 21 '11 at 11:41
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3 Answers

I recommend

Serge Lang. Differential manifolds. Addison-Wesley Publishing Co., Reading, Mass.-London- Don Mills, Ont., 1972.

Serge Lang is an excellend writer.

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Palais, "The Foundations of Global Non-linear Analysis" (or his survey article "Homotopy theory of infinite-dimensional manifolds": http://www.sciencedirect.com/science/article/pii/0040938366900024) are handy to have at hand.

EDIT: Also this paper of Eliasson might be useful: "Geometry of manifolds of maps" (1967) Journal of Differential Geometry (available at http://www.intlpress.com/JDG/archive/1967/1-1&2-169.pdf).

Of course, it's always best to see these things in action rather than in the abstract. If you know some differential geometry I can recommend Donaldson & Kronheimer "Geometry of 4-manifolds" (though much of what they do takes place in an affine Hilbert manifold, the lack of generality doesn't make the nonlinear theory significantly easier!) or McDuff & Salamon "J-holomorphic curves and symplectic topology" where they really have used Banach manifolds (for example their universal moduli spaces of pseudoholomorphic curves) and there is a lot of detail on the analysis. Another interesting setting in which infinite-dimensional analysis comes to life is the Ebin-Marsden Annals paper "Groups of diffeomorphisms and the motion of an incompressible fluid" (http://www.jstor.org/pss/1970699) where they do some Riemannian geometry (again in the Hilbert setting, I think).

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I happen to know this

Abraham, Ralph; Robbin, Joel Transversal mappings and flows. An appendix by Al Kelley W. A. Benjamin, Inc., New York-Amsterdam 1967 x+161 pp.

exists.

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