An Introduction to Analysis

When I received Jan and Piotr Mikusiński’s “An Introduction to Analysis” I sat wondering what new ideas will this book bring to me? And where will it differ from other introductory books on the subjects? There are quite a lot of them.

It didn’t take more than few pages to discover that the authors have decided to give the subject a new refreshing look.

In the first eight chapters, the authors introduce classical material with nice, short and neat proofs, examples, graphs, and above all student-teacher discussions to clarify some ideas. In chapter 1 on “Real numbers” the authors start by studying the fundamental properties of the real line. They do not construct it, but they go over properties of addition, multiplication, order and countable and uncountable subsets. Chapters 2, 3, and 4 contain a short, efficient and simple study of real functions, their limits, continuity, derivatives. Chapters 5 and 6 introduce numerical and functions sequences and series, putting special emphasis on power series, which are used in chapter 7 to build elementary functions as the unique solutions of some differential equations with initial conditions. This shows another fruitful and innovative way to look at the exponential and circular functions. Chapter 8 discusses some criteria to calculate limits: L’Hospital’s rule, Abel’s theorem on alternating series, etc. Chapter 9 studies antiderivatives and gives classical methods to calculate them.

Chapter 10 is where the book begins to differ from others. It uses absolutely convergent series to define an integral equivalent to Lebesgue’s. This chapter is by far the most difficult and abstract of all the book. Yet the authors keep their easygoing style. The results of this chapter are applied in chapter 11 to prove the fundamental theorem of calculus, to give some methods to calculate integrals, to introduce Fourier Analysis, and to discuss Euler’s gamma and beta functions.

In Chapter 12 the authors close the circle by giving a construction of the real numbers using what they call “weak equivalence”. It is rather a pity that the authors did not introduce Cauchy sequences, something they used implicitly to define real numbers.

The exercises of the book are meant to complement the text and most of them are workable. Any serious reader will do most of them.

What I liked most with this book is its style, which renders difficult notions easy to grasp by students who followed a calculus course. Yet it gives easy proofs and shows different sides of mathematical analysis. I strongly recommend it to every college library. 

Salim Salem is Professor of Mathematics at the Saint-Joseph University of Beirut.