Reference on Infinite Dimensional Manifold

Question

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.

During the 1970s (and where I was at), Werner's Foundations of Differentiable Manifolds and Lie Groups was the standard introductory text for manifolds, and those who wanted an introduction to manifolds in the infinite dimensional setting were told to look at Lang's Introduction to Differentiable Manifolds. (I see that Lang's book is now freely available on the internet.) — Dave L. Renfro, Mar 10 '14 at 15:17
Related: https://math.stackexchange.com/q/5029969/491450 , https://math.stackexchange.com/q/295358/491450 , https://math.stackexchange.com/q/2227945/491450 , https://math.stackexchange.com/q/1272537/491450 , https://math.stackexchange.com/q/4242743/491450 , https://math.stackexchange.com/q/2080992/491450 , How are infinite-dimensional manifolds most commonly treated? (MathOverflow) ... — Smiley1000, Jan 31 '25 at 16:26

score 5 · Accepted Answer · answered Mar 09 '14 at 05:34

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).

score 1 · Answer 2 · answered Jan 31 '25 at 16:52

I think, but I'm not sure, that Riemann already considered infinite dimensional manifolds under the term "infinite manifoldness".

There are many approaches to the theory of infinite dimensional manifolds. One way is to mimic the definition of a manifold but replace the modelling space by an infinite dimensional vector space. This gives to Hilbert, Banach and Frechet manifolds.

Another method is to consider the pro-completion in the category of manifolds. In particular, jet manifolds are promanifolds.

A third method is to give a direct characterisation of smoothness. The theory of diffeological, Frolicher and smooth spaces follow this line of attack.

Reference on Infinite Dimensional Manifold

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