John wrote:

No, a Banach space V is reflexive if the canonical map i:V→V ** is an isomorphism


I also don't like this sentence because it seems to use the concept of smooth Fréchet manifold to define the concept of smooth Fréchet manifold! What we really want is to define smooth functions between (open subsets of) Fréchet spaces, and then use that to define smooth Fréchet manifolds, by requiring that their transition functions be smooth.

Right, in my head I thought about the model space as example of topological manifolds, then the definition of smooth maps between them etc.

I believe the definition you're proposing is weaker (except for ℝ n, where its a theorem in Michor and Kriegl's book that it's equivalent to the usual one). I'm not 100% sure it's weaker! But it's certainly not the one people usually use in the study of Fréchet manifolds.

I don't know either, but we could present the usual definition and then this - possibly - more general one and ask this as a question in the blog post.