A Modern Introduction to Mathematical Analysis

A Modern Introduction to Mathematical Analysis, by Alessandro Fonda, is a well-written text that  provides an enjoyable narrative of the major results of (advanced undergraduate or elementary  graduate) analysis. As a text, it discusses its results in a clear voice and moves at a brisk, but not  unmanageable, place. The text does not coddle its reader, nor does it punish them by leaving its proofs  as “exercises.” In fact, there are no exercises at all – a fact that makes this manuscript very useable as a  reference but quite challenging as a classroom textbook. In the introduction, Fonda offers a list of  potential companion texts: workbooks or textbooks full of problem sets that can be worked alongside  this textbook.  

The text is self-contained, beginning with principles of logical reasoning and basic building blocks such as  set theory, the formal definition of a function, the Peano axioms, and the algebraic structure of the real  numbers as an ordered field. The introductory chapters also bring the attention of the reader to the  complex numbers, higher dimensional Euclidean space, and general metric spaces as well. In each case,  the definitions, axioms, and basic algebraic structures of the spaces are introduced, and some basic  results are proven, but no space is lingered on for too long – as a new student seeing (for example)  metric spaces for the first time, I think I would be a bit overwhelmed. In starting at this location, the text  walks a careful line of orienting the student to the principles of logic, the language of set theory, and the  most important spaces of mathematical analysis, as well as exposing readers to well-reasoned and well written expository proofs… but, at the same time, never taking on the role of “coach” in teaching its  readers how to reason logically or how to write expository proofs. Again, this leads me to believe this  text is best used as a strong reference for those already familiar with the material (or at least, very  familiar with methods of proof and logic), rather than as an elementary text for undergraduate students. 

There are two ways in which I feel this text really stands out. The first is in its introduction of the  fundamental transcendental functions. The exponential, logarithmic, trigonometric functions, among  others, are given the full treatment and are exposed and explored in a natural, easy-to-follow manner.  These functions are then returned to, again and again, where appropriate in the text. We get a full  treatment on their integrals, derivatives, Taylor series, etc., in a way that feels completely natural and  complements the theory being discussed. 

The second way in which this text distinguishes itself is in its treatment of the integral, which is  grounded in the theory of Kurzweil and Henstock. This theory is similar to that of the Riemann integral,  but uses a positive gauge function on our interval to define a point-dependent “maximal subinterval  width size” (rather than the constant mesh used for the Riemann integral). This offers a theory of  integration that generalizes the Riemann integral from the outset, and leads to easier, more robust  extensions of integration later on. (Indeed, a function is integrable in the Kurzweil-Henstock sense if and  only if it and its absolute value are integrable according to Lebesgue.) It is an approach to integration  that is well done, especially at a text of this level. 

I would not recommend this book as an introductory text for undergraduate students new to the topic,  but I would recommend it to anyone needing a refresher on the fundamental results and foundations in  Analysis, or for those looking for a different perspective on the material.

John Ross is an assistant professor of mathematics at Southwestern University.