TITLE: Fréchet manifolds vs ILH manifolds 
QUESTION [7 upvotes]: What is the precise relation between ILH manifolds and  Fréchet manifolds? Specifically:

Does any ILH manifold has a canonical structure of a Fréchet manifold?
If so, is it true that any ILH submanifold is a Fréchet sumanifold?

 Background.
The notion of an ILH (Inverse Limit Hilbert) manifold was developed by Omori in his studies of diffeomorphism groups. Very roughly it is a manifold modelled on a topological vector space that is the inverse limit of a countable family of Hilbert spaces. This object is easier to deal with than the usual Fréchet manifolds due to the avaiability of the inverse function theorem. 
For all I know the inverse limit of Hilbert spaces need not be Fréchet. More precisely, it seems that the inverse limit inherits a countable family of seminorms from the Hilbert spaces, but I see no reason for the inverse limit to be complete (as a uniform space), and I am not sure if the inverse limit is always Hausdorff.
The ILH manifold I am trying to understand is the diffeomorphism group of a compact manifold, so in particular, it is what Omori called a  strong ILH manifold, which perhaps makes a difference in answering 1-2. The diffeomorphism group is also a Fréchet manifold (indeed, it is the so called Fréchet-Lie group), but I am not sure how ILH and Fréchet manifold structures interact.

REPLY [2 votes]: Dear Igor
In order to answer your questions we first need to ensure what we mean by a Fréchet manifold. 
Obviously a Fréchet manifold should be modeled on Fréchet spaces, so the transition maps should be smooth maps between Fréchet spaces. Thats where the problem lies. 
How is differentiability of maps between Fréchet spaces defined ?
There is no natural (canonical) way to extend the classical notion of
differentiability from Banach spaces to Fréchet spaces. 
(because there is no norm in  a Fréchet space)
There are many inequivalent ways to define differentiability in
Fréchet spaces, and the choice may depend o the applications in mind. 
For some of them there are even Nash-Moser  type Inverse Function Theorems.
That was the main reason we considered the ILH structure when studying
diffeomorphism groups, or the groups of pseudo differential operators or Fourier integral operators.  
For $Diff^{\infty}(M)$ the Fréchet spaces are given as spaces of smooth sections of certain vector bundles and the Fréchet topology of  $Diff^{\infty}(M)$ is known as the smooth compact-open topology.
In other words, we don't start with a sequence of Hilbert spaces to get as
inverse limit our Fréchet space,  but we have the Fréchet space to begin with
and consider it as inverse limit of Hilbert spaces by changing the topologies.
This way we have to our proposal the whole powerful theory of Hilbert spaces.
Rudolf Schmid