Beginning functional analysis.

*(English)*Zbl 1002.46001
Undergraduate Texts in Mathematics. New York, NY: Springer. xi, 197 p. (2002).

The book under review is designed as a text for a first course on functional analysis for advanced undergraduates or for beginning graduate students. The book can be read by students who have had first courses in linear algebra and real analysis. The knowledge of Lebesgue measure and integration is not a required prerequisite for reading this book.

Chapters 1 - 5 form the core of the book. The first two chapters introduce metric spaces, normed spaces, and inner product spaces and their topology. The third chapter is devoted to Lebesgue measure and integration in \({\mathbb R}^n\). The presentation of this material is mainly concentrated on what is needed for its use in functional analysis. The fourth chapter is concerned with Fourier analysis in Hilbert spaces, drawing connections between the first two chapters and the third one. Chapter 5 introduces the reader to bounded linear operators acting on Banach spaces, Banach algebras, and spectral theory. Besides most of the standard material of these topics, the author presents an introduction to the invariant sub-space problem and a careful discussion of spectral properties of Hermitian operators. Chapter 6 is devoted to more advanced topics: Weierstrass approximation theorem, Baire category theorem, three classical theorems from functional analysis, existence of a nonmeasurable set, contraction mapping, \(C([a,b])\) as a ring and its maximal ideals, Hilbert space methods in quantum mechanics.

The text is carefully written throughout. Many exercises varying from straightforward to challenging, are useful for self-study or independent study of the book. These exercises are carefully worked out and – like exercises in all mathematical studies – indispensable (“you must do math in order to learn math”). The text contains many interesting remarks on the historical origins of mathematical concepts and theorems, and biographical information on mathematicians who contributed to the theories presented (e.g. M. FrĂ©chet, F. Riesz, P. Enflo, M. Stone). The book is highly recommendable for a first course in functional analysis.

Chapters 1 - 5 form the core of the book. The first two chapters introduce metric spaces, normed spaces, and inner product spaces and their topology. The third chapter is devoted to Lebesgue measure and integration in \({\mathbb R}^n\). The presentation of this material is mainly concentrated on what is needed for its use in functional analysis. The fourth chapter is concerned with Fourier analysis in Hilbert spaces, drawing connections between the first two chapters and the third one. Chapter 5 introduces the reader to bounded linear operators acting on Banach spaces, Banach algebras, and spectral theory. Besides most of the standard material of these topics, the author presents an introduction to the invariant sub-space problem and a careful discussion of spectral properties of Hermitian operators. Chapter 6 is devoted to more advanced topics: Weierstrass approximation theorem, Baire category theorem, three classical theorems from functional analysis, existence of a nonmeasurable set, contraction mapping, \(C([a,b])\) as a ring and its maximal ideals, Hilbert space methods in quantum mechanics.

The text is carefully written throughout. Many exercises varying from straightforward to challenging, are useful for self-study or independent study of the book. These exercises are carefully worked out and – like exercises in all mathematical studies – indispensable (“you must do math in order to learn math”). The text contains many interesting remarks on the historical origins of mathematical concepts and theorems, and biographical information on mathematicians who contributed to the theories presented (e.g. M. FrĂ©chet, F. Riesz, P. Enflo, M. Stone). The book is highly recommendable for a first course in functional analysis.

Reviewer: Joachim Naumann (Berlin)

##### MSC:

46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |

47-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory |

28-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration |