Analysis I (Q5896800)

Among the first year university students, analysis is considered as the most difficult course to study. Many abstract concepts are introduced for the first time in it. Besides, usually the most of the results are given in a highly rigorous way supplying by fairly deep proofs. Any teacher on analysis stays behind the following question: either to reduce the level of rigority in order to make the course more understandable or to follow the high standard in order to give to students a good base for the further study of mathematics. In the reviewed book the course is organized quite unusually. Great attention is paid to the questions of foundation of different aspects of analysis, as well as to the description of the main ideas. Starting very slowly with general concepts the author tries then to give more and more pieces of material to be studied by the students themselves. It is made in the form of quite deep but instructive exercises. Since that the author can follow the main stream of the course on fairly rigorous level creating the background for the study of other mathematical disciplines. The first volume consists of eight main parts of the basic calculus divided into 11 chapters and supplied by two appendices. The book starts with Preface and three introductory chapters. In the Preface the description of the whole course is presented as well as the main ideas, pedagogical and methodological aims are discussed. It is addressed mainly to teachers in analysis. Then follows Chapter 1 ``Introduction'' in which the same as in Preface is discussed but already on the beginners' (students') level. The following questions are answered: What is analysis? Why do analysis? The introductory part is continued by Chapter 2 ``The natural numbers'' and by Chapter 3 ``Set theory''. Formal and informal description of these concepts are presented up to Peano's axioms, axioms of a set and Russel's paradox. Analysis starts with the main definitions dealing with the notion of function. At last the idea of cardinal numbers is given. The next part describes numbers. It consists of two chapters, namely, Chapter 4 ``Integers and rationals'' and Chapter 5 ``The real numbers''. In fact, integer and rational numbers are introduced in somewhat axiomatic way basing on the already, by Peano's axioms, defined natural numbers. For the construction of real numbers the Cauchy sequences approach is taken summarized in the axiomatic decription. It makes possible to understand the next conceptions of analysis more easy. Two chpaters describe the idea of limit, the most crucial idea of analysis. This part starts with Chapter 6 ``Limits of sequences''. Since the notion of the Cauchy sequence is already in use, the definition of the limit is fairly straightforward. It gives possibility to introduce in Chapter 7 ``Series'' the main ideas of infinite series (of numbers) as well as to present some results on functional series. An extension of the set theory started already in Chapter 3 is continued on a higher level in Chapter 8 ``Infinite sets''. In a sense it performs prerequisites for the further study of concepts of continuity and differentiation. The main ideas of continuity are presented in Chapter 9 ``Continuous functions''. The most of classical results as of local as of global type are given here. Chapter 10 ``Differentiation of functions'' is very short but standard. Riemann integration (Chapter 11) is developed in such a way that it can be easilty compared later with other types of integration. It makes this part of the text again conceptual and highly instructive. Two appendices play a different role. The first one ``The basics of mathematical logic'' is forward-looking and serves to preserve the level of rigority. The second one ``The decimal system'' is back-looking aiming to refresh the knowledges of students about numbers which they have got in secondary school. The book in the whole is a good base for the further study of analysis and can be recommended as for teachers in Analysis for including pieces of material in the corresponding courses as for students for (self-)study of the main ideas of Analysis. For Part II (2006) see [Zbl 1108.26002].