Tim wrote:

> We call a function between manifolds smooth if it maps smooth curves to smooth curves; in this way we get the definition of a smooth Fréchet manifold.

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.