Scale calculus problems

Show that the shift action $\sigma$ of #homework 7 is a scale-smooth map by the following steps:

a) find a scale-Banach space Failed to parse (unknown function "\math"): \mathcal C = (C_\ell)_{\ell\geq 1}<\math> with <math>C_1=C^1(S^1,\mathbb R^2)<\math> b) show that <math>\sigma: S^1 \times \mathcal C \to \mathcal C<\math> is <math>sc^1<\math> and calculate <math>T\sigma<\math> c) iteratively show that <math>\sigma<\math> is <math>sc^k<\math> for all k Note: The actual map in #homework 7 is a restriction of this map here to a submanifold of <math>\mathcal C<\math> (taking values in <math>S^1<\math>, of fixed degree).