Techniques of Functional Analysis for Differential and Integral Equations

Paul Sacks’s Techniques of Functional Analysis for Differential and Integral Equations, hereon referred to as Techniques, is exactly what the title states it to be. I wish I would have had this book as a first year graduate student in applied mathematics. In fact, the book is meant to serve as a textbook for a course in analysis for beginning graduate students in applied mathematics.

Techniques begins with a motivational chapter on differential and integral equations. Here we see the basic types of problems, for example the existence of solutions to initial value problems, boundary value problems, etc., that can be efficiently treated via the methods of functional analysis. As one proceeds through the book it is observed that the examples and ideas from chapter one are a reccurring theme in examples and exercises throughout the remainder of text.

After the motivating chapter, the next several chapters develop the essential theory necessary for applying the powerful machinery of functional analysis to differential and integral equations, and also occasionally problems from numerical analysis. Topics include the basics of Banach and Hilbert space, the theory of distributions, Fourier analysis and spectral theory for linear operators. Sacks does a highly commendable job in developing the material in a way that is accessible to the audience for which the book is intended. In fact, I believe that the book could also be studied by advanced students in physics or engineering who are interested in seeing some of the theoretical underpinnings for the types of equations often arising in those fields. I think that in particular, the treatment of variational problems in chapter 15 may appeal to mathematically inclined physicists and engineers.

What I like the most about Techniques is that it takes the reader from vector spaces and metric spaces, topics familiar to undergraduate students, to concepts and methods that can be, and often are used in current research in applied mathematics. Also, while there are many other books that overlap with this one, there are few that tie the material together and connect the principal theoretical ideas with interesting applications in the way that Sacks does.

While the overwhelming majority of results cited in Techniques are proven within the text, there are some that are not. As such, the book is not entirely self-contained. However, for those results which are not accompanied by a proof, a detailed reference is given. Often, Functional Analysis by Walter Rudin is the reference of choice. In Techniques, there is a brief appendix on the Lebesgue integral and some related topics. So, measure theoretic notions are neither a prerequisite for nor developed fully, but there is enough there on the subject to use it where it is needed most. There is also a section where some basic complex analysis is used but the reader is given fair warning. I actually believe that study of Techniques provides excellent motivation for a reader to want to learn more analysis, be it real, functional, etc. Moreover, after a careful study of the book, the interested reader is well-prepared to go on to further study in functional analysis, and certainly well-prepared for further study in analysis driven applied mathematics.

For readers with interest in the theory or application of differential equations, integral equations, optimization, or numerical analysis, Techniques of Functional Analysis for Differential and Integral Equations is a very valuable resource. I highly recommend this book to any such person. I also believe that the book can serve as a nice supplement to more abstract texts on functional analysis, helping one to see how the abstract theory influences thinking about other areas of mathematics.

Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.