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The following books and/or notes develop various aspects of the theory of infinite-dimensional manifolds:

  1. Lang, Fundamentals of Differential Geometry.
  2. Kriegl & Michor, The Convenient Setting of Global Analysis.
  3. Choquet-Bruhat & DeWitt-Morette, Analysis, Manifolds and Physics.
  4. Klingenberg, Riemannian Geometry.
  5. Marsden, Ratiu, and Abraham, Manifolds, Tensor Analysis, and Applications.
  6. Hamilton, The inverse function theorem of Nash and Moser.

Question: Are there any other books that systematically develop from scratch the theory of infinite-dimensional manifolds, in particular Frechet manifolds?

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2 Answers 2

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It's not specifically a text on infinite-dimensional manifolds, but the theory of diffeology developed in Patrick Iglesias-Zemmour's text encompasses infinite dimensional manifolds of all sorts (Banach, Frechet, etc.). The category of diffeological spaces is also a quasi-topos, which is a really fantastic property.

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Infinite-Dimensional Lie Groups by Karl-Hermann Neeb looks really nice. It includes basics of the theory of infinite-dimensional manifolds in chapter 2. He also shows how the theory of linear ODEs collapses for Frechet spaces.

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