TITLE: Intuition for failure of Implicit Function theorem on Frechet Manifolds
QUESTION [28 upvotes]: When dealing with moduli spaces of, say connections or metrics, I am using the notions of Frechet spaces/manifolds/groups. I have become familiar with Banach manifolds (I think), but Frechet manifolds less so. In my studies, we want to apply the Inverse and Implicit Function Theorems, which work on Banach spaces, but apparently fail for Frechet spaces (and as a result we pass to Sobolev completions).
Why does it fail (intuitively), or why should I expect it to fail? What is the hinge that allows it to work for Banach but not Frechet? Do I have to do a lot more to get an analogous result in the land of Frechet? This should also help me see the differences between a Banach and Frechet manifold, because right now it seems subtle.

REPLY [6 votes]: In a Banach setting the proof of the Implicit Function Theorem (IFT) is based on Newton iteration. Each iteration step requires the solution of a linearized equation. When applying this, for example, to a partial differential equation $F(u)=f$, problems arise, even if the base manifold is compact, if the linearizations $DF(u)$ (assumed invertible) are not elliptic. The reason is "loss of derivatives" in the iterative corrections $DF(u)^{-1}F(u)$. Application by $F$ decreases (Sobolev or Hölder) differentiability by the order of $F$, and this loss is regained by $DF(u)^{-1}$ only if $DF(u)$ is elliptic. If $DF(u)$ is not elliptic, then the techniques of Nash and Moser are used to obtain an IFT. Here the iteration scheme is modified by mollification: In each iteration step $u$ is replaced by $S_\theta u$ for a suitable mollification parameter $\theta$, $S_\theta u\to u$ in each Hölder space as $\theta\to\infty$. It is essential that norm estimates such as
$$\|S_\theta u\|_\alpha\lesssim\theta^{\alpha-\beta}\|u\|_\beta,\quad\alpha\geq\beta,$$
allow control of higher order derivatives of the mollifications of $u$ by lower order derivatives of $u$. (It would be of no help to assume $u\in C^\infty$ from the beginning because this gives no control of higher order derivatives.) The method of Nash was simplified and applied by Hörmander; see chapter 3 of the book of Alinhac and Gerard (GSM 82) for an exposition and references. Hamilton abstracted the estimates as tameness properties and gave a generalization of the IFT to Frechet spaces. 
So, my intuition about IFT in a Frechet setting comes from the special, but significant, situation which I just described: The non-ellipticity of the derivative/linearization of the differential operator and the resulting loss in derivatives during iteration are the cause of trouble. A remedy if is approximation by suitable mollications.