Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) which exposes the theory completely in the setting of smooth sections, that is using Fréchet space techniques.

In particular, I'm interested in how the Fredholm theory translates into Fréchet spaces. The basic results should transfer without big modifications, however some results are not so straightforward (like "Fredholm operators are open in the space of all continuous linear maps" since the theory of dual spaces is more complicated).

  • $\begingroup$ Just want to say that this is an excellent question! $\endgroup$ – Igor Khavkine Apr 6 '14 at 14:09
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    $\begingroup$ Perhaps Treves's Locally convex spaces and linear partial differential equations (1967) is written in that spirit. However, it does not appear to treat Fredholm theory. $\endgroup$ – Igor Khavkine Apr 6 '14 at 17:10
  • $\begingroup$ Thank you for this reference, looks indeed like a good start. In addition, the work "The inverse function theorem of Nash and Moser" of Hamilton also treatises differential operators and even a big part of the elliptic theory (discusses for example greens functions). More literature recommendations are still highly welcomed. $\endgroup$ – Tobias Diez Apr 6 '14 at 19:45

The book Locally Convex Spaces by Osborne has a statement on p152 of the Fredholm alternative for Hausdorff locally convex spaces. Also consider the exercises from 26 onwards of the same chapter.

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