I originally posted this question on StackExchange, where it was suggested I post here. It was also suggested I read about Hilbert manifolds and Fréchet manifolds. Nevertheless, I am still looking for an answer to (mainly the first part of) my question.
At a summer school I recently attended, infinite-dimensional manifolds popped up. I have never worked with them before (although I'm very familiar with finite-dimensional manifolds). The lecturer at the school did not give any details about the technical realization of infinite-dimensional manifolds, mentioning that there were issues (such as picking a topology) that he would leave out for the sake of clarity, since the relevant results were true independent of the exact technical details. An internet search reveals that Banach manifolds are one way of treating infinite-dimensional manifolds, but there are others.
Are Banach manifolds the most common way of defining infinite-dimensional manifolds, or are there other notions commonly used? Is there a more or less universal consensus about when to use which treatment? What are the most important (dis)advantages of each?
Supposing I want to learn the basics of infinite-dimensional manifolds, are there any well-written introductory texts you would recommend? (on StackExchange, The Convenient Setting of Global Analysis by A. Kriegl and P. Michor was recommended)
Best Answer
Banach manifolds have found many uses such as, gauge theory (Donaldson theory, Seiberg-Witten theory, Floer theory), symplectic topology (Gromov-Witten theory), to name few I am more familiar with.
One great advantage of Banach manifolds over Frechet manifolds is the implicit function theorem which in the Banach context takes a simpler form, and thus Banach manifolds are easier to recognize.
One disadvantage of Banach manifolds over Frechet manifolds is the fact that natural notions of real analyticity are harder to implement on Banach spaces.
One place to learn about Banach manifolds is Lang's Differential and Riemannian Manifolds