Functional analysis. An introductory course (Q1650055)

This textbook is well organized and the proofs are carefully written. A brief description of the contents of the book is as follows: Chapter 1 is devoted to preliminaries. Chapter 2 is dedicated to metric spaces and their completions. Important properties of topological vector spaces, some classical sequence spaces, and spaces of continuous functions are presented in Chapter 3. In Chapter 4, the author deals with normed spaces and operators between them. The interrelation of compactness of the closed balls and the finite dimensionality of normed spaces is discussed in this chapter. A study of linear functionals and the Hahn-Banach theorem is given in Chapter 5. Chapter 6 is concerned with the other ``classical fundamental theorems'' of functional analysis, that is, the uniform boundedness theorem, the open mapping theorem, and the closed graph theorem. Finally, Hilbert spaces, orthogonality, orthonormal sets, and types of Hilbert space operators are discussed in Chapter 7. The reviewer thinks that it would be nice if the next edition of the book would consider the operator norm on the space of square complex matrices and then provide several observations such as Example 7.19 of the book. In addition, it would be interesting if the weak topology on normed spaces (and Hilbert spaces) were introduced and the elementary properties of this topology were described. Each chapter is concluded with an interesting note and several exercises, helping the reader to better understand the topics of the chapter. The natural prerequisites for the book are linear algebra and real analysis. Providing many examples, it will be useful for upper-undergraduate and beginning graduate students.