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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?

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The canonical reference is "The Convenient Setting of Global Analysis" by A. Kriegl and P. Michor. Available here

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  • $\begingroup$ Thank you. Any ideas about the first part of the question? $\endgroup$ – Daan Michiels Feb 9 '13 at 11:54
  • $\begingroup$ @Daan I only have some superficial acquaintance with this topic, sadly. Anyway, searches on "Hilbert Manifold" and "Fréchet_manifold" give some other directions to thought. Maybe (a rephrased version of) this question deserves asking on MathOVerflow where Peter Michor is seen frequently... $\endgroup$ – Yuri Vyatkin Feb 9 '13 at 12:17
  • $\begingroup$ Ok, thanks. I'll post on MathOverflow. $\endgroup$ – Daan Michiels Feb 9 '13 at 13:31
  • $\begingroup$ I have posted the question on MathOverflow at mathoverflow.net/questions/121306/…. $\endgroup$ – Daan Michiels Feb 9 '13 at 13:43
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Regarding Fréchet manifolds, while it is rather terse, the best book I know is:

V. V. Sharko, Functions on manifolds, Translations of Mathematical Monographs, Volume 131, American Mathematical Society, Providence, RI (1993). Algebraic and topological aspects, Translated from the Russian by V. V. Minachin.

For diffeological spaces, the canonical reference is P. Iglessias-Zemmour's upcoming book (former preprint) Diffeology.

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