Answer by Rudolf Schmid for Fréchet manifolds vs ILH manifolds

2011-09-08T20:34:33Z

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

User rudolf schmid - MathOverflow

Answer by Rudolf Schmid for Fréchet manifolds vs ILH manifolds