A Companion to Analysis. A second First and First Second Course in Analysis
This interesting book is intended "for those students who might find rigorous analysis a treat". They should read it after the first (undergraduate) course or before the second (graduate) course in analysis. It is assumed that the reader has a basic knowledge of linear algebra and has experience with the use and manipulation of limits. The book is not a usual systematic textbook. The author mainly explains the hardest problems, which must be resolved in order to obtain a rigorous development of the calculus, and easier facts are not mentioned or are left to the reader as exercises. The author starts with a discussion of real numbers; he emphasizes how important completeness of the real line is. Then, almost all standard topics of differential calculus of one and more variables are discussed. The theory of the one-dimensional Riemann integral is described in detail. Improper integrals, integrals of two variables and the Riemann-Stieltjes integrals are briefly discussed. The basic theory of complete metric spaces with applications to differential equations is also presented. Moreover, some more special advanced topics are discussed in an interesting elementary way. Among them are: Shannon‘s theorem, geodesics, first steps in the calculus of variations, Green’s functions for ordinary differential equations of second order, and the idea of a constructive analysis. Each chapter contains a number of exercises directly related to the exposition. A further 345 exercises (whose solutions are electronically available) are contained in the appendix; many of them lead the reader through standard pieces of theory. The book is written in a very personal style and contains many witty comments, quotations and historical remarks. The exposition is very precise, but it is also interesting and enthusiastic. The book can be warmly recommended to gifted and hardworking students and could also be useful for teachers of analysis.