Tame Fréchet space

Summary

A tame Fréchet space is a graded Fréchet space of a special kind. Up to tame linear isomorphism, it is a direct summand of a space of exponentially decreasing sequences in a Banach space.

The notion is due to Richard Hamilton. Tame spaces carry smoothing operators, and this is what the Nash–Moser theorem needs to run Newton's method with loss of derivatives. The basic example is the space of smooth functions on a compact manifold, graded by the $C^n$ norms.