##
**A course in abstract analysis.**
*(English)*
Zbl 1258.28004

Graduate Studies in Mathematics 141. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9083-7/hbk). xv, 367 p. (2012).

In the course of the past forty years, the author has published several excellent textbooks in analysis. His standard texts “Functions of one complex variable” [Graduate Texts in Mathematics. 11. New York-Heidelberg-Berlin; Springer- Verlag. XI, 313 p. (1973; Zbl 0277.30001)], “Functions of one complex variable. II” [Graduate Texts in Mathematics. 159. New York, NY: Springer-Verlag. xvi, 394 p. (1995; Zbl 0887.30003)], “A course in functional analysis. 2nd ed.” [Graduate Texts in Mathematics, 96. New York etc.: Springer-Verlag. xvi, 399 p. DM 148.00 (1990; Zbl 0706.46003)], and “A course in operator theory” [Graduate Studies in Mathematics. 21. Providence, RI: American Mathematical Society (AMS). xv, 372 p. (2000; Zbl 0936.47001)] are very popular and widely used worldwide, due to the author’s outstanding expository mastery.

The book under review, the most recent one among his various primers on advanced analysis, grew out of the author’s two-semester course in abstract analysis that he taught during the academic year 2010/2011. Having been designed to prepare first-year graduate students for the PhD Qualifying example, this course included a semester of measure theory followed by a semester of basic functional analysis. With regard to this special project, the author has set a high value on simplifying and streamlining the presentation of measure theory, and on formulating the contents of functional analysis so they depend only on the measure theory appearing in the same book, thereby providing a self-contained treatment of these two subjects at the level and depth appropriate for beginning graduate students. This particular didactic strategy is verbatim explained in the preface, along with the author’s other guiding principles of teaching advanced mathematics.

As to the precise contents, the text consists of eleven chapters, each of which is subdivided into several sections.

Chapter 1 is titled “Setting the stage” and contains a mix of basic topics needed to study the following parts of the book. The reader meets here Riemann-Stieltjes integrals, a little of metric space theory, basics on normed vector spaces, locally compact spaces, and the concept of linear functional. Chapter 2 introduces the elements of measure theory, with particular emphasis on the Radon measure space, measurable functions, integration with respect to a measure, Fatou’s lemma and the classical convergence theorems, signed measures, and the basics on \(L^p\)-spaces.

Chapter 3 gives a first introduction to Hilbert spaces, with the idea to present the Riesz representation theorem for later use in measure theory. In fact, Chapter 4 returns to measure theory and completes the treatment given in the present book. The main topics of this chapter are as follows: the Lebesgue-Radon-Nikodým Theorem, complex functions and measures, linear functionals on \(L^p\)-spaces, functions of bounded variation, product measures, the Lebesgue measure on \(\mathbb{R}^d\), differentiation on \(\mathbb{R}^d\), absolutely continuous functions, convolutions, and the Fourier transform. Chapter 5 begins the study of functional analysis and is, like the following chapters, based on the author’s earlier textbook “A course in functional analysis” [(1990; Zbl 0706.46003)]. This chapter is primarily devoted to linear transformations (or operators), first on Banach spaces but quickly focusing on Hilbert spaces, thus continuing the discussion begun in Chapter 3. Sesquilibear forms, adjoint operators, compact linear operators, and the spectral theorem for compact Hermitean operators are among the main topics touched upon in this chapter. Banach spaces are more comprehensively studied in Chapter 6, with particular emphasis on the Hahn-Banach Theorem, the Open Mapping Theorem, the Closed Graph Theorem, and the Banach-Steinhaus Theorem.

Chapter 7 provides the very basic facts on locally convex spaces, but only to the extent needed to develop duality theory in functional analysis. This includes metrizable locally convex spaces as well as some geometric consequences of the Hahn-Banach Theorem. Duality theory itself is the theme of Chapter 8, with special focus on the relation between a Banach space and its dual space, including the Krein-Milman Theorem and the Stone-Weierstrass Theorem.

More generally, various (weak) topologies on a locally convex space \(X\) and its dual space \(X^*\) are explored and applied and the properties of reflexive Banach spaces are discussed, too. Chapter 9 returns to the study of linear transformations between Banach spaces, thereby extending some of the results given in Chapter 5 to this more general setting. Compact operators, Schauder’s Theorem, the Arzelà-Ascoli Theorem, and, related results are explained in this short chapter.

Chapter 10 presents the foundations of the theory of Banach algebras over the field of complex numbers and describes the spectral theory of (compact) operators on a Banach space. This presentation leads to the Fredholm Alternative, on the one hand, and to the Gelfand-Mazur Theorem in the context of Abelian Banach algebras on the other. Chapter 11 provides an introduction to \(C^*\)-algebras. Based on the groundwork laid in the previous chapter, this final part develops elementary properties and examples of \(C^*\)-algebras, turns then to a functional calculus for normal operators on a Hilbert space, and concludes with constructing a complete set of unitary invariants for a normal operator on a separable Hubert space.

There are two appendices complementing the main text. One of them concerns the Baire Category Theorem for complete metric spaces, while the other one gathers a sequence of definitions and propositions from the theory of nets in topology, as they were used in Chapter 7. In general, each section in the book comes with a list of related exercises complementing the main text, and the whole book is interlarded with a wealth of illustrating, highly instructive examples.

Altogether, this textbook on measure theory and functional analysis reflects the author’s rich teaching experience just as evidently as his pedagogical skills.The utmost lucid presentation is fully self-contained with regard to first-year graduate students, and the author’s strategy of introducing new material gradually, thereby starting with the particular and working up to the general, appears to be particularly efficient and user-friendly. No doubt, the author’s newest textbook is an excellent primer on measure theory and functional analysis, and a perfect companion to the venerable classical text “Foundations of modern analysis” by A. Friedman [New York etc.: Holt, Rinehart and Winston, Inc. VI, 250 p. (1970; Zbl 0198.07601)] likewise.

The book under review, the most recent one among his various primers on advanced analysis, grew out of the author’s two-semester course in abstract analysis that he taught during the academic year 2010/2011. Having been designed to prepare first-year graduate students for the PhD Qualifying example, this course included a semester of measure theory followed by a semester of basic functional analysis. With regard to this special project, the author has set a high value on simplifying and streamlining the presentation of measure theory, and on formulating the contents of functional analysis so they depend only on the measure theory appearing in the same book, thereby providing a self-contained treatment of these two subjects at the level and depth appropriate for beginning graduate students. This particular didactic strategy is verbatim explained in the preface, along with the author’s other guiding principles of teaching advanced mathematics.

As to the precise contents, the text consists of eleven chapters, each of which is subdivided into several sections.

Chapter 1 is titled “Setting the stage” and contains a mix of basic topics needed to study the following parts of the book. The reader meets here Riemann-Stieltjes integrals, a little of metric space theory, basics on normed vector spaces, locally compact spaces, and the concept of linear functional. Chapter 2 introduces the elements of measure theory, with particular emphasis on the Radon measure space, measurable functions, integration with respect to a measure, Fatou’s lemma and the classical convergence theorems, signed measures, and the basics on \(L^p\)-spaces.

Chapter 3 gives a first introduction to Hilbert spaces, with the idea to present the Riesz representation theorem for later use in measure theory. In fact, Chapter 4 returns to measure theory and completes the treatment given in the present book. The main topics of this chapter are as follows: the Lebesgue-Radon-Nikodým Theorem, complex functions and measures, linear functionals on \(L^p\)-spaces, functions of bounded variation, product measures, the Lebesgue measure on \(\mathbb{R}^d\), differentiation on \(\mathbb{R}^d\), absolutely continuous functions, convolutions, and the Fourier transform. Chapter 5 begins the study of functional analysis and is, like the following chapters, based on the author’s earlier textbook “A course in functional analysis” [(1990; Zbl 0706.46003)]. This chapter is primarily devoted to linear transformations (or operators), first on Banach spaces but quickly focusing on Hilbert spaces, thus continuing the discussion begun in Chapter 3. Sesquilibear forms, adjoint operators, compact linear operators, and the spectral theorem for compact Hermitean operators are among the main topics touched upon in this chapter. Banach spaces are more comprehensively studied in Chapter 6, with particular emphasis on the Hahn-Banach Theorem, the Open Mapping Theorem, the Closed Graph Theorem, and the Banach-Steinhaus Theorem.

Chapter 7 provides the very basic facts on locally convex spaces, but only to the extent needed to develop duality theory in functional analysis. This includes metrizable locally convex spaces as well as some geometric consequences of the Hahn-Banach Theorem. Duality theory itself is the theme of Chapter 8, with special focus on the relation between a Banach space and its dual space, including the Krein-Milman Theorem and the Stone-Weierstrass Theorem.

More generally, various (weak) topologies on a locally convex space \(X\) and its dual space \(X^*\) are explored and applied and the properties of reflexive Banach spaces are discussed, too. Chapter 9 returns to the study of linear transformations between Banach spaces, thereby extending some of the results given in Chapter 5 to this more general setting. Compact operators, Schauder’s Theorem, the Arzelà-Ascoli Theorem, and, related results are explained in this short chapter.

Chapter 10 presents the foundations of the theory of Banach algebras over the field of complex numbers and describes the spectral theory of (compact) operators on a Banach space. This presentation leads to the Fredholm Alternative, on the one hand, and to the Gelfand-Mazur Theorem in the context of Abelian Banach algebras on the other. Chapter 11 provides an introduction to \(C^*\)-algebras. Based on the groundwork laid in the previous chapter, this final part develops elementary properties and examples of \(C^*\)-algebras, turns then to a functional calculus for normal operators on a Hilbert space, and concludes with constructing a complete set of unitary invariants for a normal operator on a separable Hubert space.

There are two appendices complementing the main text. One of them concerns the Baire Category Theorem for complete metric spaces, while the other one gathers a sequence of definitions and propositions from the theory of nets in topology, as they were used in Chapter 7. In general, each section in the book comes with a list of related exercises complementing the main text, and the whole book is interlarded with a wealth of illustrating, highly instructive examples.

Altogether, this textbook on measure theory and functional analysis reflects the author’s rich teaching experience just as evidently as his pedagogical skills.The utmost lucid presentation is fully self-contained with regard to first-year graduate students, and the author’s strategy of introducing new material gradually, thereby starting with the particular and working up to the general, appears to be particularly efficient and user-friendly. No doubt, the author’s newest textbook is an excellent primer on measure theory and functional analysis, and a perfect companion to the venerable classical text “Foundations of modern analysis” by A. Friedman [New York etc.: Holt, Rinehart and Winston, Inc. VI, 250 p. (1970; Zbl 0198.07601)] likewise.

Reviewer: Werner Kleinert (Berlin)

### MSC:

28Axx | Classical measure theory |

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

47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |

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

28C05 | Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures |

46A03 | General theory of locally convex spaces |

46A20 | Duality theory for topological vector spaces |

46Bxx | Normed linear spaces and Banach spaces; Banach lattices |

46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |

46J10 | Banach algebras of continuous functions, function algebras |

46L05 | General theory of \(C^*\)-algebras |

47A10 | Spectrum, resolvent |