Mathematical Preliminaries

Abstract

In this chapter, for the convenience of the reader, we recall some basic mathematical notions and theorems that we shall often use in the following chapters. To a number of important theorems and lemmas we provide proofs, especially when they are straightforward and simple enough or when we find them very instructive. We also clarify some notions, providing examples, special cases, and exercises. We believe that the best illustration of the few abstract theorems from functional analysis stated in this chapter, that may seem too abstract at the first reading, is provided in the subsequent chapters through their direct application to concrete problems in partial differential equations of hydrodynamics. Thus, if a theorem seems too abstract, we advice the reader to skip it without becoming discouraged and then concentrate on Section 1 in Chapter 2, which contains fundamental material for this part of the book.