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).

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