I am studying manifold. For comprehension, I read the site http://en.wikipedia.org/wiki/Manifold, and there is some information about infinite dimensional manifold.
Now I have two questions or requests:
(1) When was infinite dimensional manifold introduced? I guess this may be related to Functional Analysis. But I want more details about its history.
(2) I am still curious about the properties about infinite dimensional manifold, especially local same as topological vector space. Could someone give a reference about it. Thanks.
The book "Riemannian Geometry" by Wilhelm Klingenberg does include an infinite-dimensional setting from the start, if I remember correctly, that is, your manifold is modelled on any separable Hilbert or Banach space (for a Riemannian metric, you obviously need a Hilbert space though).
The most extensive treatment that I know if is the Book "The convenient setting of global analysis" by Peter Michor, that treats manifolds modeled on any locally convex vector space. This book has a lot a stuff and a lot of functional analysis in particular.
Regarding the first question, I don't know a definite answer, but I think that infinite-dimensional manifolds where considered first in Physics, where the appear somewhat naturally (even though they may not have been called that way or were given a mathematically rigoros treatment).