Preliminaries

In this chapter we review some preliminary material from Calculus and Functional Analysis and also recall the basic properties of real analytic functions. We introduce the notion of Roumieu classes, then summarize the classical properties of Hölder continuous functions. Next, we introduce the notion of coordinate cylinders and of sets that are hypographs of Cartesian functions locally around the boundary points, including the case of Lipschitz sets. Finally, we turn to Schauder spaces and the corresponding embedding theorems and to the subsets of ℝ n $${\mathbb {R}}^n$$ of class C m, α. Although this part of the book is mostly self-contained, we sometimes refer to other textbooks for basic results of Calculus and of Real and Functional Analysis.