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Functional Analysis: A Gentle Introduction, Volume 1

Dzung Minh Ha
Matrix Editions
Publication Date: 
Number of Pages: 
[Reviewed by
Susan Slattery
, on

Having never taken or taught a course in functional analysis, I thought I would be well-suited to reviewing a book subtitled “A Gentle Introduction.” In fact, the author makes a point of stating that only a working knowledge of linear algebra is needed — even better since my own real analysis days are long past! I approached the book with an open mind, trying to see the book both as an instructor looking to select a textbook and as an advanced student preparing to take a first course in functional analysis.

In the Preface, the author indicates that there is enough material for a two semester course, and provides suggestions on selecting certain chapters and sections for several different one-semester courses. The content of the book seems fairly typical, as judged by comparison with several other introductory functional analysis textbooks. The proofs are complete and provide sufficient detail to avoid the common student complaint that the author “skipped too many steps.” There are numerous examples and exercises in each section.

As I worked through the text, however, I began to suspect that students who had not taken an introductory course in real analysis might have trouble with the proofs and understanding the significance of some of the results. A quick look back at the author’s suggestions in the Preface confirmed my suspicions: the more significant results are in sections in which the author recommends a background in real analysis, including some point-set topology. In fact, the author indicates that, for students with only calculus and linear algebra, a one-semester course would consist of only Chapters 0 and 1 together with Section 5.1. This content reduces the functional analysis course to, essentially, an introduction to metric and topological spaces, and barely scratches the surface of the content of the book. There are certainly more appropriate books for teaching metric and topological spaces!

On a technical note, every significant result, including certain off-set lines in proofs and examples, are assigned line numbers for reference. Unfortunately, numbers are duplicated in some places so, for example, the number “2.3.4” is assigned to both a line in a proof and a definition. The line numbering scheme is also used within the statements of propositions, theorems and definitions, in addition to any “1), 2), 3)” listing that occurs. Consequently, the numbers in some sections get quite large, and the reader must be careful to note whether it is Corollary 4.5.4 or line (4.5.4) they should be referencing. The cover of the book says “Volume 1,” so I hope the author fixes this numbering scheme in later volumes.

Overall, I found the book to be an acceptable introduction to functional analysis in spite of the issues raised above and I would consider this book if I were selecting texts for a course. I would, however, strongly advise that students ignore the author’s “only a working knowledge of linear algebra” comment and complete a real analysis course before attempting to read this book.

Susan Slattery teaches at Villa Julie College in Stevenson, MD.

1.1 Metrics and metric spaces
1.2 Open and closed sets
1.3 Topological spaces
1.4 Continuous functions
1.5 Open sets and continuity
1.6 Some important topological concepts
1.7 Convergence of sequences in metric spaces
1.8 Completeness
1.9 Density, separability, and approximation
1.10 Metric space completions
1.11 Compactness
1.12 The Banach fixed point theorem
1.13 Baire's category theorem

Chapter 2: Normed spaces 116

2.1 Linear operators on function spaces
2.2 Hamel bases
2.3 Norms and normed spaces
2.4 Topological concepts in normed spaces
2.5 Topological vector spaces
2.6 Kolmogorov's theorem
2.7 Banach spaces
2.8 Infinite series in normed spaces
2.9 Schauder bases
2.10 Linear functionals and hyperplanes
2.11 Constructing new normed spaces

Chapter 3: Operators on normed spaces 206

3.1 Continuous linear maps
3.2 Integral operators
3.3 Linear homeomorphisms
3.4 Three important theorems
3.5 The normed space B(X,Y)
3.6 Complementary subspaces and projections
3.7 Riesz's lemma
3.8 The spectrum of a bounded linear operator
3.9 Continuous linear functionals and dual spaces

Chapter 4: Inner product spaces 291

4.1 Definitions and examples
4.2 Orthogonality
4.3 Unitary isomorphisms
4.4 Inner product spaces: three problems
4.5 Three characterizations for Hilbert spaces
4.6 Hilbert bases

Chapter 5: The Banach space C(X) 365

5.1 The Arzela-Ascoli theorem
5.2 Korovkin's theorem and the Weierstrass approximation theorem
5.3 Sub-algebras
5.4 The Stone-Weierstrass theorem

Chapter 6: Additional topics 415

6.1 The Baire-Osgood theorem
6.2 Gram determinants and Muntz's theorem
6.3 Differential equations

Appendix A: Set theory and functions 464

A.1 Sets
A.2 Relations
A.3 Zorn's lemma and the axiom of choice
A.4 Functions
A.5 Cardinality
A.6 The axiom of completeness on R

Appendix B: Mostly linear algebra (a brief review) 480

B.1 Polynomials and sequences
B.2 Vector spaces
B.3 Linear independence and span
B.4 Bases and dimension
B.5 Linear transformations
B.6 Partial derivatives and the mean value theorem
B.7 Riemann integrals

Appendix C: Some technical results 491

Appendix D: Solutions to odd-numbered exercises 494

Bibliography 618

Notation 622

Index 624


akirak's picture

As the publisher, I would like to clarify the numbering used in the book, which apparently was not adequately explained. All theorems, definitions, propositions, etc. are referred to explicitly, with numbers not in parentheses: for instance, "proposition 2.11.5". Equation numbers or numbers referring to lines in equations are given in parentheses, with no identifier. For instance, "if (3.4.33) holds" refers to equation 3.4.33. Propositions, definitions, theorems, lemmas, etc. all share the same numbering system, for instance in section 3.1. we find, in order, proposition 3.1.1, definition 3.1.2, theorem 3.1.3, definition 3.1.4, proposition 3.1.5, and not proposition 3.1.1, definition 3.1.1, theorem 3.1.1, definition 3.1.2, proposition 3.1.2. Thus the numbers always increase, which (in theory) should make results easier to find.