This book provides a thorough introduction to (linear) functional analysis and operator theory in Banach and Hilbert spaces. There are no prerequisites except a course in advanced calculus and, for some chapters, an elementary knowledge of complex function theory. No measure theory is employed or mentioned. Only rudimentary topological or algebraic concepts are used. The value of the book is enhanced by the fact that each chapter ends with a section “Problems”, which will not be listed in the following table of contents:
Chapter 1. Basic Notions (1.1 A problem from differential equations, 1.2 An examination of the results, 1.3 Examples of Banach spaces, 1.4 Fourier series).
Chapter 2. Duality (2.1 The Riesz representation theorem, 2.2 The Hahn-Banach theorem, 2.3 Consequences of the Hahn-Banach theorem, 2.4 Examples of dual spaces).
Chapter 3. Linear Operators (3.1 Basic properties, 3.2 The adjoint operator, 3.3 Annihilators, 3.4 The inverse operator, 3.5 Operators with closed ranges, 3.6 The uniform boundedness principle, 3.7 The open mapping theorem).
Chapter 4. The Riesz theory for compact operators (4.1 A type of integral equation, 4.2 Operators of finite rank, 4.3 Compact operators, 4.4 The adjoint of a compact operator).
Chapter 5. Fredholm operators (5.1 Orientation, 5.2 Further properties, 5.3 Perturbation theory, 5.4 The adjoint operator, 5.5 A special case, 5.6 Semi-Fredholm operators, 5.7 Products of operators).
Chapter 6. Spectral theory (6.1 The spectrum and resolvent sets, 6.2 The spectral mapping theorem, 6.3 Operational calculus, 6.4 Spectral projections, 6.5 Complexification, 6.6 The complex Hahn-Banach theorem, 6.7 A geometric lemma).
Chapter 7. Unbounded operators (7.1 Unbounded Fredholm operators, 7.2 Further properties, 7.3 Operators with closed ranges, 7.4 Total subsets, 7.5 The essential spectrum, 7.6 Unbounded semi-Fredholm operators, 7.7 The adjoint of a product of operators).
Chapter 8. Reflexive Banach spaces (8.1 Properties of reflexive spaces, 8.2 Saturated subspaces, 8.3 Separable spaces, 8.4 Weak convergence, 8.5 Examples, 8.6 Completing a normed vector space).
Chapter 9. Banach algebras (9.1 Introduction, 9.2 An example, 9.3 Commutative algebras, 9.4 Properties of maximal ideals, 9.5 Partially ordered sets, 9.6 Riesz operators, 9.7 Fredholm perturbations, 9.8 Semi-Fredholm perturbations, 9.9 Remarks).
Chapter 10. Semigroups (10.1 A differential equation, 10.2 Uniqueness, 10.3 Unbounded operators, 10.4 The infinitesimal generator, 10.5 An approximation theorem).
Chapter 11. Hilbert space (11.1 When is a Banach space a Hilbert space?, 11.2 Normal operators, 11.3 Approximation by operators of finite rank, 11.4 Integral operators, 11.5 Hyponormal operators).
Chapter 12. Bilinear forms (12.1 The numerical range, 12.2 The associated operator, 12.3 Symmetric forms, 12.4 Closed forms, 12.5 Closed extensions, 12.6 Closable operators, 12.7 Some proofs, 12.8 Some representation theorems, 12.9 Dissipative operators, 12.10 The case of a line or a strip, 12.11 Selfadjoint extensions).
Chapter 13. Selfadjoint operators (13.1 Orthogonal projections, 13.2 Square roots of operators, 13.3 A decomposition of operators, 13.4 Spectral resolution, 13.5 Some consequences, 13.6 Unbounded selfadjoint operators).
Chapter 14. Measures of operators (14.1 A seminorm, 14.2 Perturbation classes, 14.3 Related measures, 14.4 Measures of noncompactness, 14.5 The quotient space, 14.6 Strictly singular operators, 14.7 Norm perturbations, 14.8 Perturbation functions, 14.9 Factored perturbation functions).
Chapter 15. Examples and applications (15.1 A few remarks, 15.2 A differential operator, 15.3 Does A have a closed extension?, 15.4 The closure of A, 15.5 Another approach, 15.6 The Fourier transform, 15.7 Multiplication by a function, 15.8 More general operators, 15.9 \(B\)-Compactness, 15.10 The adjoint of \(\overline A\), 15.11 An integral operator).
This is an excellent book e.g. for somebody working in applied mathematics who wants to learn operator theory from scratch. It contains a wealth of material on (semi-) Fredholm operators and perturbations. The material is presented in a very elegant way, starting with motivations (for instance, see Chapter 1, sections 4.1, 10.1 etc.). The approach is substantially different than in most other mathematics books: As the author wrote in the introduction to the first edition, “After introducing the first topic, I try to follow a trend of thought wherever it may lead without stopping to fill in details. I do not try to describe a subject fully at the place it is introduced. Instead, I continue with my trend of thought until further information is needed. Then I introduce the required concept or theorem and continue with the discussion.” E.g., in 2.2 only the real case of the Hahn-Banach theorem is given since the author believes that the reader will see the advantage of complex Banach spaces only in Chapter 6. In 6.6 the complex form of the Hahn-Banach theorem is stated, and in 7.3 its geometric form is discussed. Zorn’s lemma appears in 9.5 for the first time. Rather often the author continues to explain his trend of thought, postponing (part of the) proofs. He certainly does this in a masterly fashion, and thus the book is very pleasant to read: Sometimes there are imaginary conversations between the author and the reader like after Theorem 6.8: “Hold a minute!” you exclaim. “You are using results for a complex Banach space that were proved only for real Banach spaces.” Yes, you are right. However,... (and so on).
Since there are practically no prerequisites, the book does not contain certain topics treated by many other textbooks on functional analysis. Fréchet spaces or locally convex spaces are not mentioned, only weak convergence of sequences occurs, but no weak topology. Consequently, only the elementary part of Gelfand theory is included in 9.3. The notion of \(C^*\)-algebra is never defined. \(L^p\)-spaces are used only over intervals or the real line. The Arzelà-Ascoli theorem and the Stone-Weierstrass theorem are not given. But which other textbook on functional analysis would contain so many results on (semi-) Fredholm operators and their perturbation?
The book finishes with a long glossary and a compilation of the major theorems, but relatively few references. The author mentions in the text that there are compact operators on Banach spaces which cannot be approximated by finite rank operators, but that is all the information on this topic which the reader is given. Enflo’s name is not mentioned, and the bibliography does not contain any of his papers. There are no historical notes, and a remark like the one at the end of 2.4 (“Theorem 2.14 is due to F. Riesz”) is very rare. For somebody interested in historical remarks and perspectives as well as references to related material, the German textbook “Funktionalanalysis” by Dirk Werner is recommended (1995; Zbl 0831.46002).