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


I recommend
Serge Lang is an excellend writer. 


Palais, "The Foundations of Global Nonlinear Analysis" (or his survey article "Homotopy theory of infinitedimensional 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/1967/11&2169.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 4manifolds" (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 "Jholomorphic 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 infinitedimensional analysis comes to life is the EbinMarsden 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). 


I happen to know this Abraham, Ralph; Robbin, Joel Transversal mappings and flows. An appendix by Al Kelley W. A. Benjamin, Inc., New YorkAmsterdam 1967 x+161 pp. exists. 

