Elementary Functional Analysis / Graduate Texts in Mathematics Bd.253 (PDF)

Everything is proved completely, at a pretty high but not uncomfortable pace, and, as already indicated, the flow of material is excellent. What’s more, each chapter is followed by a long list of exercises that should afford the reader the opportunity to take it all to the next level, from absorbing the presentation (take notes, fill the margin of the book with comments, doodle… whatever floats your boat…) to doing battle on one’s own, so to speak (although some problems come equipped with hints).

So it is that in the span of about 200 pages, MacCluer takes the reader from Hilbert space preliminaries to Banach- and C*-algebras and, of course, the spectral theorem (introduced by a quote from Paul Halmos stressing that it all begins with Hermitian matrices). The book’s preface indicates that MacCluer’s intent is to present an effective introduction to the "huge field" of functional analysis, suitable for self-study, with modest preliminary requirements placed on the reader. She achieves her goal beautifully: Elementary Functional Analysis is a wonderful book.

-Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.


“MacCluer (Univ. of Virginia), right from the opening epigraph (from Hadamard), acknowledges the strangeness of functional analysis. … provide intellectual context, miraculously manages to develop in detail just what a student must learn to continue in a grounded way in quite a number of subjects: advanced functional analysis, single operator theory, Banach space geometry, and Banach or operator algebras. Summing Up: Highly recommended. Upper-division undergraduate through professional collections.” (D. V. Feldman, Choice, Vol. 47 (3), November, 2009)

“Books on functional analysis are not exactly ubiquitous, but every publishing house has several text books in this area on its backlist. MacCluer’s book is an excellent new entry. … What distinguishes this book from many others is its reader-friendly style that is opposite to the dry definition-theorem-proof format found in many other expositions. … Each chapter is accompanied by a section of well-chosen, highly interesting and often non-canonical exercises … . It is a pleasure to read and highly recommended.” (Dirk Werner, Mathematical Reviews, Issue 2010 b)

“The present textbook is a concise introduction to linear functional analysis. … What I like about the book is its user-friendly style, the many historic remarks and comments mingled with the main text, and the wealth of nice exercises. … In summary, it is a nice contribution to the existing literature on this subject.” (G. Teschl, Internationale Mathematische Nachrichten, Issue 213, April, 2010)

“This interesting book covers material from the basic elements of functional analysis to more advanced topics such as spectral theory and is intended for specialists and graduate students in mathematics and other disciplines. … Each chapter includes some historical commentaries and many examples which make the book readable as well as a lot of exercises which invite the reader on a long trip to the ‘functional analysis land’. The book consists of … six chapters, a bibliography containing 48 references and subject index.” (Mohammad Sal Moslehian, Zentralblatt MATH, Vol. 1170, 2009)

“The book being reviewed is written mainly for graduate students, although much of the material can also be studied by final year students. … There is an abundance of examples to illustrate the concepts involved. … The presentation of material is excellent: elegant, precise and concise; everything is completely proved at a reasonably comfortable pace. … It is a beautiful and wonderful small book to be studied, treasured and kept for reference.” (Peter Shiu, The Mathematical Gazette, Vol. 94 (531), November, 2010)