Preliminaries

In this chapter, we provide the main notation and facts from real and functional analysis. We introduce the spaces of smooth functions, the Sobolev spaces, and the spaces of divergence-free vector functions, which play a fundamental role in studying the Navier–Stokes equations. Additionally, we describe some results from general topology, geometric measure theory, and harmonic analysis that are used in this book. Finally, we formulate some basic properties of solutions to elliptic equations. Our aim is not to present an exhaustive treatment of the auxiliary results. Most of these facts are given without proof, and for more detailed study, we provide references to specialized literature. However, we do present proofs of selected assertions from geometric measure theory and harmonic analysis, which, in our opinion, are not as commonly found in the classical theory of differential equations. We have endeavored to present proofs that are not overly complicated, but the choice of statements to be proved depends on our preferences.