### The preface to “A First Course in Functional Analysis”

I am not yet done being excited about my new book, A First Course in Functional Analysis. I will use my blog to advertise my book, one last time. This post is for all the people who might wonder: “why did you think that anybody needs a new book on functional analysis?” Good question! The answer is contained in the preface to the book, which is pasted below the fold.

#### 1. In a nutshell

The purpose of this book is to serve as the accompanying text for a first course in functional analysis, taken typically by second- and third-year undergraduate students majoring in mathematics. As I prepared for my first time teaching such a course, I found nothing among the countless excellent textbooks in functional analysis available that perfectly suited my needs. I ended up writing my own lecture notes, which evolved into this book (an earlier version appeared on my blog).

The main goals of the course this book is designed to serve are to introduce the student to key notions in functional analysis (complete normed spaces, bounded operators, compact operators), alongside significant applications, with a special emphasis on the Hilbert space setting. The emphasis on Hilbert spaces allows for a rapid development of several topics: Fourier series and the Fourier transform, as well as the spectral theorem for compact normal operators on a Hilbert space.

I did not try to give a comprehensive treatment of the subject, the opposite is true. I did my best to arrange the material in a coherent and effective way, leaving large portions of the theory for a later course. The students who finish this course will be ready (and hopefully, eager) for further study in functional analysis and operator theory, and will have at their disposal a set of tools and a state of mind that may come in handy in any mathematical endeavor they embark on.

The text is written for a reader who is either an undergraduate student, or the instructor in a particular kind of undergraduate course on functional analysis. The background required from the undergraduate student taking this course is minimal: basic linear algebra, calculus up to Riemann integration, and some acquaintance with topological and metric spaces (in fact, the basics of metric spaces will suffice; and all the required material in topology/metric spaces is collected in the appendix).

Some “mathematical maturity” is also assumed. This means that the readers are expected to be able to fill in some details here and there, not freak out when bumping into a slight abuse of notation, and so forth. (For example, a “mathematically mature” reader needs no explanation as to what mathematical maturity is :-).

#### 2. More details on the contents and on some choices made

This book is tailor-made to accompany the course Introduction to Functional Analysis given at the Technion — Israel Institute of Technology. The official syllabus of the course is roughly: basic notions of Hilbert spaces and Banach spaces, bounded operators, Fourier series and the Fourier transform, the Stone-Weierstrass theorem, the spectral theorem for compact normal operators on a Hilbert space, and some applications. A key objective, not less important than the particular theorems taught, is to convey some underlying principles of modern analysis.

The design was influenced mainly by the official syllabus, but I also took into account the relative place of the course within the curriculum. The background that I could assume (mentioned above) did not include courses on Lebesgue integration or complex analysis. Another thing to keep in mind was that besides this course, there was no other course in the mathematics undergraduate curriculum giving a rigorous treatment of Fourier series or the Fourier transform. I therefore had to give these topics a respectable place in class. Finally, I also wanted to keep in mind that students who will continue on to graduate studies in analysis will take the department’s graduate course on functional analysis, in which the Hahn-Banach theorems and the consequences of Baire’s theorem are treated thoroughly.

This allowed me to omit these classical topics with a clean conscience, and use my limited time for a deeper study in the context of Hilbert spaces (weak convergence, inverse mapping theorem, spectral theorem for compact normal operators), including some significant applications (PDEs, Hilbert functions spaces, Pick interpolation, the mean ergodic theorem, integral equations, functional equations, Fourier series and the Fourier transform).

An experienced and alert reader might have recognized the inherent pitfall in the plan: how can one give a serious treatment of $L^2$ spaces, and in particular the theory of Fourier series and the Fourier transform, without using the Lebesgue integral? This is a problem which many instructors of introductory functional analysis face, and there are several solutions which can be adopted.

In some departments, the problem is eliminated altogether, either by making a course on Lebesgue integration a prerequisite to a course on functional analysis, or by keeping the introductory course on functional analysis free of $L^p$ spaces, with the main examples of Banach spaces being sequence spaces or spaces of continuous functions. I personally do not like either of these easy solutions. A more pragmatic solution is to use the Lebesgue integral as much as is needed, and to compensate for the students’ background by either giving a crash course on Lebesgue integration or by waving one’s hands where the going gets tough.

I chose a different approach: hit the problem head on using the tools available in basic functional analysis. I define the space $L^2[a,b]$ to be the completion of the space of piecewise continuous functions on $[a,b]$ equipped with the norm $\|f\|_2 = (\int_a^b |f(t)|^2 dt)^{1/2}$, which is defined in terms of the familiar Riemann integral. We can then use the Hilbert space framework to derive analytic results, such as convergence of Fourier series of elements in $L^2[a,b]$, and in particular we can get results on Fourier series for honest functions, such as $L^2$ convergence for piecewise continuous functions, or uniform convergence for periodic and $C^1$ functions.

Working in this fashion may seem clumsy when one is already used to working with the Lebesgue integral, but, for many applications to analysis it suffices. Moreover, it shows some of the advantages of taking a functional analytic point of view.

I did not invent the approach of defining $L^p$ spaces as completions of certain space of nice functions, but I think that this book is unique in the extent to which the author really adheres to this approach: once the spaces are defined this way, we never look back, and everything is done with no measure theory.

To illustrate, in Section 8.2 we prove the mean ergodic theorem. A measure preserving composition operator on $L^2[0,1]$ is defined first on the dense subspace of continuous functions, and then extended by continuity to the completion. The mean ergodic theorem is proved by Hilbert space methods, as a nice application of some basic operator theory. The statement (see Theorem 8.2.5) in itself is significant and interesting even for piecewise continuous functions — one does not need to know the traditional definition of $L^2$ in order to appreciate it.

Needless to say, this approach was taken because of pedagogical constraints, and I encourage all my students to take a course on measure theory if they are serious about mathematics, especially if they are interested in functional analysis. The disadvantages of the approach we take to $L^2$ spaces are highlighted whenever we stare them in the face; for example, in Section 5.3, where we obtain the existence of weak solutions to PDEs in the plane, but fall short of showing that weak solutions are (in some cases) solutions in the classical sense.

The choice of topics and their order was also influenced by my personal teaching philosophy. For example, Hilbert spaces and operators on them are studied before Banach spaces and operators on them. The reasons for this are (a) I wanted to get to significant applications to analysis quickly, and (b) I do not think that there is a point in introducing greater generality before one can prove significant results in that generality. This is surely not the most efficient way to present the material, but there are plenty of other books giving elegant and efficient presentations, and I had no intention — nor any hope — of outdoing them.

#### 3. How to use this book

A realistic plan for teaching this course in the format given at the Technion (13 weeks, three hours of lectures and one hour of exercises every week) is to use the material in this book, in the order it appears, from Chapter 1 up to Chapter 12, skipping Chapters 6 and 11. In such a course, there is often time to include a section or two from Chapters 6 or 11, as additional illustrative applications of the theory. Going through the chapters in the order they appear, skipping chapters or sections that are marked by an asterisk, gives more or less the version of the course that I taught.

In an undergraduate program where there is a serious course on harmonic analysis, one may prefer to skip most of the parts on Fourier analysis (except $L^2$ convergence of Fourier series), and use the rest of the book as a basis for the course, either giving more time for the applications, or by teaching the material in Chapter 13 on the Hahn-Banach theorems. I view the chapter on the Hahn-Banach theorems as the first chapter in further studies in functional analysis. In the course that I taught, this topic was given as supplementary reading to highly motivated and capable students.

There are exercises spread throughout the text, which the students are expected to work out. These exercises play an integral part in the development of the material. Additional exercises appear at the end of every chapter. I recommend for the student, as well as the teacher, to read the additional exercises, because some of them contain interesting material that is good to know (e.g., Gibbs phenomenon, von Neumann’s inequality, Hilbert-Schmidt operators). The teaching assistant will also find among the exercises some material better suited for tutorials (e.g., the solution of the heat equation, or the diagonalization of the Fourier transform).

There is no solutions manual, but I invite any instructor who uses this book to teach a course, to contact me if there is an exercise that they cannot solve. With time I may gradually compile a collection of solutions to the most difficult problems.

Some of the questions are original, most of them are not. Having been a student and a teacher in functional and harmonic analysis for several years, I have already seen many similar problems appearing in many places, and some problems are so natural to ask that it does not seem appropriate to try to trace who deserves credit for “inventing” them. I only give reference to questions that I deliberately “borrowed” in the process of preparing this book. The same goes for the body of the material: most of it is standard, and I see no need to cite every mathematician involved; however, if a certain reference influenced my exposition, credit is given.

The appendix contains all the material from metric and topological spaces that is used in this book. Every once in while a serious student — typically majoring in physics or electrical engineering — comes and asks if he or she can take this course without having taken a course on metric spaces. The answer is: yes, if you work through the appendix, there should be no problem.

There are countless good introductory texts on functional analysis and operator theory, and the bibliography contains a healthy sample. As a student and later as a teacher of functional analysis, I especially enjoyed and was influenced by the books by Gohberg and Goldberg, Devito, Kadison and Ringrose, Douglas, Riesz and Sz.-Nagy, Rudin, Arveson, Reed and Simon, and Lax. These are all recommended, but only the first two are appropriate for a beginner. As a service to the reader, let me mention three more recent elementary introductions to functional analysis, by MacCluer, Hasse, and Eidelman-Milman-Tsolomitis. Each one of these looks like an excellent choice for a textbook to accompany a first course.

I want to acknowledge that while working on the book I also made extensive use of the Web (mostly Wikipedia, but also MathOverflow/StackExchange) as a handy reference, to make sure I got things right, e.g., verify that I am using commonly accepted terminology, find optimal phrasing of a problem, etc.

#### 5. Acknowledgments

This book could not have been written without the support, encouragement and good advice of my beloved wife, Nohar. Together with Nohar, I feel exceptionally lucky and thankful for our dear children: Anna, Tama, Gev, Em, Shem, Asher and Sarah.

I owe thanks to many people for reading first drafts of these notes and giving me feedback. Among them are Alon Gonen, Shlomi Gover, Ameer Kassis, Amichai Lampert, Eliahu Levy, Daniel Markiewicz, Simeon Reich, Eli Shamovich, Yotam Shapira, and Baruch Solel. I am sorry that I do not remember the names of all the students who pointed a mistake here or there, but I do wish to thank them all. Shlomi Gover and Guy Salomon also contributed a number of exercises. A special thank you goes to Michael Cwikel, Benjamin Passer, Daniel Reem and Guy Salomon, who have read large portions of the notes, found mistakes, and gave me numerous and detailed suggestions on how to improve the presentation.

I bet that after all the corrections by friends and students, there are still some errors here and there. Dear reader: if you find a mistake, please let me know about it! I will maintain a page on my personal website in which I will collect corrections.

I am grateful to Sarfraz Khan from CRC Press for contacting me and inviting me to write a book. I wish to thank Sarfraz, together with Michele Dimont the project editor, for being so helpful and kind throughout. I also owe many thanks to Samar Haddad the proofreader, whose meticulous work greatly improved the text.

My love for the subject and my point of view on it were strongly shaped by my teachers, and in particular by Boris Paneah (my Master’s thesis advisor) and Baruch Solel (my Ph.D. thesis advisor). If this book is any good, then these men deserve much credit.

My parents, Malka and Meir Shalit, have raised me to be a man of books. This one, my first, is dedicated to them.