This chapter presents some key elements of real and functional analysis employed throughout this book. The scope of this chapter is not to provide a comprehensive discussion on this topic, which is widely covered by a plethora of well-referenced books. The purpose of this chapter is to present all foundational elements of this book succinctly, coherently, and rigorously. This chapter can be studied as a first step toward the main goal of this book, namely the design of control systems and machine learning methods based on reproducing kernel Hilbert spaces (native spaces), or can be consulted by the reader as a compendium of resources for a deeper understanding of the material presented in later chapters. Numerous references are provided throughout this chapter to guide the reader to a deeper understanding of the topics discussed herein.