In this chapter provides a brief introduction to the basic concepts and theorems of metric spaces, vector spaces, normed spaces, Banach spaces, inner product spaces, and Hilbert spaces, which are necessary to explain the subject of this book, the Aleksandrov–Rassias problems. For this purpose, we mainly refer to the book (Debnath and Mikusiński, Introduction to Hilbert Spaces with Applications, 2nd edn. Academic, New York, 2005) by L. Debnath and P. Mikusiński, among others. In Sect. 1.6, we will systematically prove, from the point of view of a basic knowledge level, that \(\ell ^2\) is a real Hilbert space. For this end, we mainly refer to the book by R. H. Kasriel (Undergraduate Topology. W. B. Saunders, Philadelphia, 1971). Readers who are familiar with the mathematical objects mentioned above can skip this chapter.